An X-ray scattering study of ordering in block copolymers

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An X-ray scattering study of ordering in block copolymers
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Thesis (Ph. D.)--University of Florida, 1990.
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Includes bibliographical references (leaves 241-248).
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by Curtis Ray Harkless.
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AN X-RAY SCATTERING STUDY OF ORDERING IN BLOCK COPOLYMERS


By

CURTIS RAY HARKLESS














A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1990











ACKNOWLEDGMENTS


Firstly, I would like to express my gratitude to my advisor, Stephen E. Nagler.

His assistance in the preparation of this dissertation is greatly appreciated, and without

his supervision this work would not have been possible.

I would like to express my thanks to the beamline staff, Brian Stephenson and

Jean Jordan-Sweet, for their valuable technical assistance and to DARPA and DOE for

financial support in the early and later stages of this work, respectively.

Much of this work could not have been performed without the outstanding

technical support offered by Ward Ruby. His creativity and resourcefulness were second

only to his jokes of questionable taste.

I thank Paul Lyman and Liz Seiberling not only for their helpful discussions and

honest opinions, but also for late night support and indefatigable good humor that I could

always count on.

I would like to thank Robert F. Shannon, Jr., for all of his help. Writing this

dissertation was made a great deal easier by his company and the knowledge that

someone else was suffering as much as I.

The assistance given to me by Marsha A. Singh in all aspects of the performance

of these experiments cannot be overstated. Our trip to Brookhaven to take the data, upon

which this dissertation is based, was made less arduous by her companionship and

experimental expertise. In addition, her attention to detail was invaluable during the

writing of this dissertation.

I thank the Lord for supporting me through this challenging time.









Finally, I thank my family. The advice offered by my parents: Les and Nancy

(like "Get a job." ) and the stimulating questions posed by my sisters: Lisa, Kim, Jen,

and Stefanie (like "Aren't you done yet?" ) were greatly appreciated. But most of all, I

am grateful for their unfailing support in all of my endeavors.












TABLE OF CONTENTS




ACKNOWLEDGEMENTS.............................. ....................................................ii

LIST OF FIGU RES .............................................................................................. vii

ABSTRACT......................................... .............................................................. xii

CHAPTERS

1 INTRODUCTION.............................................................................. 1

2 PHASE TRANSITION KINETICS................................ ........... .... 9
Phase Transition Kinetics--General Concepts ...................................... 9
Nucleation Theory............................................................................ 17
Spinodal Decomposition ................................................................ 21
Domain Growth ........................................................................ 23
Transformation Curves........................................................... ..... 24
Late Stage Growth--Coarsening .................................... ............ 32
Universal Classification..................................................... .......... 34

3 X-RAY DIFFRACTION............................................... ............ 36
Fundamentals of X-ray Diffraction................................. .......... 36
Small Angle X-ray Scattering............................................................. 41
Scattering From Liquids............................................... ............ 47
Scattering From Ordered Structures .............................................. 50
The Effect of Finite Size...................... .......... ........... ... 52
The Crystal Structure Factor............................. ........... .. 53
The Effect of Particle Motion .............................................. 54
The Effect of Strain ............................................................ 55
The Powder Pattern ........................................... ........... ... 56

4 REVIEW OF BLOCK COPOLYMER LITERATURE...................... 59
Polymers--General Background.................................... ........... .... 59
The Glass and Melting Transitions................................................. 61
Phase Separation in Polymer Blends................................ ............ 62
Block Copolymers--Bulk Properties.................................................... 64
Block Copolymers--Kinetic Studies .................................................... 69







Block Copolymer Solutions.......................................................... 70
Equilibrium Measurements ................................................... 72
Time-Dependent Experiments............................................ 75

5 THE EXPERIMENTAL APPARATUS ............................................ 79
System Overview ........................................................................... 79
The Polymer X-ray Scattering Furnace............................................ 82
The Furnace Body .............................................................. 82
The Translational Stage ...................................................... 85
The Sample Mount ............................................................. 86
The Quench Technique............................................................ 89
The X-ray Scattering Apparatus ..................................................... 92
The National Synchrotron Light Source.................................. 92
The Collimation System ....................................................... 94
The pre-hutch collimation system............................... 96
The in-hutch collimation system................................. 97
The Detection System .......................................................... 98
The detector operating principle................................. 98
The dark count ......................................................... 99
Preliminary Experimental Procedures............................ .............. 100
Sample Preparation.............. .............................. ........ 100
X-ray Beam Alignment and Collimation.............. ........... 105
Normalization for Incident Intensity ................................. 108
Measurement of the Parasitic Scattering ............... ............... 109
Equilibrium Data Acquisition Procedure .................................... 112
Kinetic Data Acquisition Procedure ................................................... 113

6 EQUILIBRIUM MEASUREMENTS: ANALYSIS
AND RESULTS.................................................................... 117
Presentation of the Static Data................................ 117
Identification of the X-ray Scattering Features .............................. 130
Preliminary Discussion of the Static Data................................ 131
The Effects of Experimental Resolution........................ ........ .. 133
Determination of the Structural Parameters ................................. 134
The Microdomain Parameters................................. ... 134
Analysis of the Macrolattice Structure and Dimensions ....... 137
The background contribution............................ 139
The amorphous contribution........................ 139
The crystalline contribution..................... ........ 140
Discussion of the Macrolattice Characterization..................... 140
Finite Size and Strain Effects............................................... 147
The Ordered Volume Fraction.................. ............. 148
Discussion of the Fine Structure in the SAXS Profiles......... 149







Concentration and Temperature Dependence
of the Structure............................................. ............................... 152
Behavior of the Crystalline Component ................................. 154
Behavior of the Amorphous Component.............................. 161


7 KINETIC MEASUREMENTS: ANALYSIS AND RESULTS ........ 168
Preliminary Discussion...................................................................... 168
The Ordering Kinetics................................................................... 174
The Master Curve............................................................. 193
The Crossover Behavior ...................................................... 198
Shallow Quench Oscillatory Behavior ................................... 200

8 SUMMARY AND CONCLUSIONS................................................. 206

APPENDICES

A DATA ACQUISITION PROGRAMS ............................................... 209

B THE HEATING CIRCUIT .......................................................... 227

C THE SOLENOID INTERFACE .................................................. 229

D THE DEVELOPMENT OF THE SAMPLE MOUNT....................... 231

REFERENCES .................................................................................................. ... 241

BIOGRAPHICAL SKETCH ............................................................................... 249














LIST OF FIGURES


Figure

1-1 The diblock copolymer molecule and an illustration of
microphase separation........................................... .............................. 3

1-2 The three domain morphologies commonly observed in diblock
copolymer system s...................................................................................... 5

1-3 An illustration of the ordering of spherical microdomains onto
a cubic lattice......................................................................................................... 6

2-1 An example of a phase diagram for a binary alloy exhibiting a phase
transition....................................................................................... ............ 10

2-2 The shape of the free energy curve versus order parameter for various
temperatures, above and below the transition point......................................... 13

2-3 An illustration of the different temporal regimes in the transformation
process............................................................. ................................. 16

2-4 A contrast of the nucleation and spinodal decomposition transformation
mechanisms ................................................................................................. 22

2-5 A simulated transformation curve showing the induction time and the
half-completion time........................................................................... 26

2-6 An ideal Master Curve for a transformation which obeys Cahn's
relation for nucleation on two dimensional nucleation sites............................ 31

3-1 An illustration of diffraction by two scattering centers and the definition
of the scattering angle 20 ............................................................................. 38

3-2 The scattering form factors for ideal spheres having radii: 80, 100,
and 120 A .......................................................................................... ......... 42







3-3 A plot showing the effect of polydispersity on the scattering form factor
for a gaussian distribution of spherical particles having a width parameter,
= 1, 10, and 20 A ........................................................................................ 44

3-4 The modifications to the scattered profile which result from spheres
having imperfect boundaries................................................................. 46

3-5 Model scattering profiles are shown for systems of spheres interacting
via the hard core potential at varying particle concentrations, v4/v0 ..........................49

3-6 An illustration of Bragg scattering from a crystal lattice ................................. 51

5-1 A schematic of the data acquisition system............................................... 80

5-2 A top view of the polymer x-ray scattering furnace illustrating the
relative placement of its various features as described in the text ................. 83

5-3 The final sample mount design is illustrated in a fron and top view ............. 87

5-4 A typical quench profile illustrating the base and sample temperatures as a
function of time following the quench ........................................................... 90

5-5 A schematic of the SAXS collimation system............................................ 95

5-6 The chemical formulas of polystyrene and polybutadiene along with
the selective solvent used in these studies, n-tetradecane............................. 101

5-7 A graph of the beam height (out of scattering plane ) and width
(in plane) profiles ................................................................................. 107

5-8 A graph of the measured parasitic scattering................................................. 110

6-1 Raw scattering profiles for the SB15 sample upon cooling.......................... 118

6-2 Raw scattering profiles for the SB15 sample upon heating.......................... 119

6-3 Raw scattering profiles for the SB25 sample upon cooling.......................... 120

6-4 Raw scattering profiles for the SB25 sample upon heating.......................... 121

6-5 Raw scattering profiles for the SB35 sample upon cooling.......................... 122

6-6 Raw scattering profiles for the SB35 sample upon heating.......................... 123

6-7 Raw scattering profiles for the SB50 sample upon cooling........................ 124







6-8 Raw scattering profiles for the SB50 sample upon heating........................... 125

6-9 Raw scattering profiles for the SBS25 sample upon cooling ......................... 126

6-10 Raw scattering profiles for the SBS25 sample upon heating........................ 127

6-11 Raw scattering profiles for the SBS35 sample upon cooling ....................... 128

6-12 Raw scattering profiles for the SBS35 sample upon heating........................ 129

6-13 A plot of the scattered profile for the SB50 sample at 44.0 C
illustrating the estimated spherical form factor............................................ 135

6-14 A plot showing the three components of the parameterization of
the low Q range of the scattering spectra..................................................... 138

6-15 A comparison of the simple cubic and body-centered cubic structure
specific fits to the scattered spectrum for the SB35 sample at 44.0 C......... 144

6-16 An illustration of the splitting of the first-order Bragg reflection
in the SB15 solution at 62.5 C................................................................ 150

6-17 Plots of the microdomains radius and lattice constants versus
polymer concentration at approximately 44 C ......................................... 153

6-18 Graphs of the normalized peak intensities of the primary Bragg peak vs.
temperature for each of the four polymer solutions..................................... 155

6-19 Graphs of the measured Bragg widths vs. temperature for the four
polym er solutions ......................................................................................... 158

6-20 Graphs of the measured lattice constants vs. temperature for the four
polymer solutions ................................................................................... 160

6-21 Variation of the amorphous peak amplitude vs. temperature for the
SB15 solution ............................................................................................... 162

6-22 Plots of the amorphous peak position and width vs. temperature for the
SB50 solution ......................................................................................... 164

6-23 The measured phase diagram for the polystyrene-polybutadiene /C14
solutions ................................................................................................. 166

7-1 A plot of the final spectrum as a function of quench depth for the
SB15 solution ............................................................................................... 170







7-2 A plot of the final spectrum as a function of quench depth for the
SB25 solution ............................................................................................... 171

7-3 A plot of the final spectrum as a function of quench depth for the
SB35 solution ............................................................................................... 172

7-4 A plot of the final spectrum as a function of quench depth for the
SB50 solution ............................................................................................... 169

7-5 A plot of the raw SAXS spectra as a function of time following
a quench from 159.6 C to 112.2 C on the SB25 solution............................ 175

7-6 A plot of the raw SAXS spectra as a function of time following
a quench from 130.6 C to 68.5 C on the SB15 solution............................ 177

7-7 A plot of the raw SAXS spectra as a function of time following
a quench from 159.6 C to 94.0 C on the SB25 solution............................ 178

7-8 A plot of the raw SAXS spectra as a function of time following
a quench from 170.0 C to 86.5 C on the SB35 solution............................ 179

7-9 A plot of the raw SAXS spectra as a function of time following
a quench from 201.6 C to 98.5 C on the SB50 solution............................ 180

7-10 Graphs of the peak maximum as a function of time following a
shallow quench ( upper graph ) and a quench to below the
ordering temperature ( lower graph )........................................................... 182

7-11 A 3-D plot of the temporal peak development of the scattered profile........... 183

7-12 Graphs of the fitted amorphous peak position and FWHM along
with the Bragg FWHM and the measured lattice constant vs. time
following a quench from 159.6 C to 94.0 C on the SB25
solution............................................. .................................................... 185

7-13 A plot of the normalized Bragg and amorphous fitted amplitudes
as a function of time following a quench from 159.6 OC to
94.0 C on the SB25 solution.................................................................. 188

7-14 A plot of the normalized Bragg fitted amplitudes as a function of
time following quenches from 160 oC to 102.2 OC, 94.0 oC,
82.0 oC, and 58.7 oC on the SB35 solution ................................................. 192

7-15 A plot of the quantity: In ( In ( ( 1 C )-1 ) vs. the natural
log of the time following quenches from 160 C to 102.2 OC,







94.0 C, 82.0 C, and 58.7 C on the SB35 solution...................................... 194

7-16 A master curve of all of the quench data on all of the samples
as described in the text.................................................................................. 196

7-17 The time constants, t, for the ordering transition for the SB15,
SB25, SB35, and SB50 solutions are plotted as a function of
quench depth................................................................................................. 197

7-18 A schematic illustrating the proposed origin of the observed
crossover in ordering behavior.......................................................................... 199

7-19 Graphs of the fitted and normalized Bragg amplitudes as a function
of time following quenches from ~ 202 C to 152.0 oC, 143.5 OC,
and 137.5 OC on the SB50 solution ............................................................. 201

7-20 Graphs of the raw data from a quench from 201.8 C to 152.0 C
on the SB50 solution at various times following the quench.......................... 202

7-21 Plots of the fitted Bragg widths and positions along with the
normalized intensity, Z, as a function of time following the
quench from 201.8 OC to 152.0 C on the SB50 solution............................. 204

B-l A schematic of the heating circuit........................................................... 228

C-1 A schematic of the solenoid interface............................................................ 230

D-1 An illustration of the initial sample mount design....................................... 232












Abstract of Dissertation Presented
to the Graduate School of the University of Florida
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy

AN X-RAY SCATTERING STUDY OF ORDERING IN BLOCK COPOLYMERS

By

Curtis Ray Harkless

December 1990

Chairman: Stephen E. Nagler
Major Department: Physics


The block copolymer, a novel system for studying the kinetics of first-order phase

transitions, is investigated. Solutions of the block copolymer polystyrene-polybutadiene

exhibit two types of phase transitions presently of great interest to the science

community. Studies of the process by which these transformations occur can broaden

our understanding of kinetic phenomena and aid in the identification of universal features

such as nonequilibrium scaling. This thesis represents the first attempt to probe the

kinetics of these transitions using synchrotron x-ray diffraction.

The block copolymer molecule is composed of two different polymer chains

joined by a covalent bond. When the chains are incompatible mesophases form through

the process of microphase separation. The system also exhibits an ordering transition

which results in a characteristic superlattice of the microdomains. A brief discussion of

first-order phase transition kinetics is given followed by a detailed review of the relevant

literature on block copolymers.








High quality diblock and triblock copolymer solutions were prepared. The

structure of each system was determined from the x-ray scattering profiles as a function

of temperature after which kinetic measurements were performed. Each kinetic

measurement involved annealing the sample above the dissolution temperature and

rapidly quenching the sample temperature to a fixed point below. The subsequent

transformation process was observed through the x-ray scattering profile.

Due to the resolution obtained at the synchrotron, the scattering contributions

from the ordered and disordered states are identified and separated for the first time. As

a result several new features are observed such as the presence of fine structure in the

x-ray scattering profile. Fast kinetic measurements reveal that transformation occurs as a

two-stage process and that the ordering transition exhibits an unexpected crossover in

behavior consistent with two dimensional nucleation. In addition, the velocities of

transformation, as determined from the kinetic data, follow trends expected from

fundamental thermodynamic considerations. Finally, novel oscillatory growth of the

Bragg component is observed in the shallow quench limit.














CHAPTER 1
INTRODUCTION



One of the most fascinating and active fields in statistical physics is the study of

phase transition kinetics.1 Specifically, there is much interest in understanding the non-

linear and cooperative phenomena involved in first-order phase transitions (FOPTs).2

Although instances of FOPTs in nature are abundant, much of the essential physics of

these processes remains unclear. Examples of these phenomena include phase separation

in binary alloys,3 45, 6 and binary fluids, ordering in alloys,7. 8, 9.10.11,12.13 ferromagnetic

and antiferromagnetic systems,14,15 and melting/crystallization transitions.16 One of the

foremost efforts at this time is the identification of relevant parameters to form a

universal classification scheme for these systems.17 Consequently, there is a need to

study more diverse systems and polymer systems are and ideal and novel system for this

purpose.1

A polymer is a long chain molecule composed of a number of repeat units called

monomers. Generally, each monomer itself is an organic molecule joined to adjacent

molecules by a covalent bond. Common examples of polymers include polystyrene,

polyisoprene, and poly (vinyl chloride). The excellent mechanical properties of these

materials lend themselves to a variety of industrial applications. The interesting

statistical physics involved in understanding these materials has attracted the attention of

many notable physicists including de Gennes, Debye,18 Kramers,19 and Flory. Excellent

references on the static and dynamic properties of polymers have been written by Doi

and Edwards,20 Flory,21 and de Gennes.22









The block copolymer molecule is formed by joining two chemically distinct

polymer chains end to end with a covalent bond and is the subject of the experimental

studies to be presented here. A typical block copolymer is illustrated in Figure 1-1.

Much recent work has been done toward understanding these complicated systems.23

When the monomers, A and B, are incompatible ( A-A B-B, nearest neighbor

configurations are favored over A-B configurations ), the system attempts to minimize

internal energy by forming A- or B-rich domains thereby reducing the surface to volume

ratio between A and B regions. This type of arrangement leads to a loss of

conformational entropy, resulting in an equilibrium structure that is a balance between

these two opposing forces. At high temperatures, the entropic force dominates and the

system is a homogeneous melt as shown in Figure 1-1. As the temperature is lowered,
the balance shifts to formation of domains at the dissolution temperature, Td. This

domain formation is similar to the phase separation observed in binary alloys. However,

the junction of A and B monomers in the block copolymer limits the ultimate size of the

phase separated regions to microscopic dimensions. This process is appropriately termed

microphase separation and results in the different domain morphologies observed in these

materials.24

Phase separation in binary systems has been well studied and has led to the

formulation of very general relations describing the phase transition kinetics in conserved

order parameter systems.' Microphase separation is an intriguing analog of this process

involving both the complex process of polymer interdiffusion and an inherent constraint

to the size of the phase separated regions. For this reason, block copolymers are useful

systems for broadening our knowledge of kinetic phenomena and testing the limitations

of present models.

A variety of domain morphologies has been observed in block copolymer systems

through x-ray scattering23 and transmission electron microscopy2.26 and the type of

morphology was found to depend on the fractional composition of the individual block














THE BLOCK COPOLYMER MOLECULE

AAAA----AAAABBBB----BBBB






MICROPHASE SEPARATION


T > Td










T < Td


Figure 1-1 The diblock copolymer molecule and an illustration of microphase
separation.








copolymer chains. For example, polymers having a small A block relative to the B block

are known to form spherical A-rich microdomains in a B-rich matrix. Remarkably, these

systems are seen to exhibit long-range order in the form of a cubic lattice of the spherical

microdomains. The specific type of cubic structure is presently the subject of some

debate27 and, for this reason, structure determination is discussed in detail in this work.

In those copolymers where the A block is somewhat smaller than the B block, a

2-dimensional hexagonal closed packed ( HCP ) arrangement of cylindrical domains is

formed. A lamellar structure results if the A and B blocks are of comparable size.

These features are summarized in Figure 1-2.

Often in studying these materials, it is useful to employ a solvent which

selectively dissolves one block of the copolymer. In this way the interaction potential

between unlike monomers is partially screened resulting in a lower, experimentally

accessible dissolution temperature. An added benefit of using selective solvents is the

ability to control the effective fractional composition of the chain by selectively swelling

one block relative to the other. The study of these polymer solutions is also important

from an industrial standpoint in that additives are commonly incorporated in pure

polymer materials to improve their already advantageous mechanical properties.

Further interesting effects were observed in small angle x-ray scattering (SAXS)

studies of polystyrene(PS)-polybutadiene (PB) diblock copolymer in the selective solvent

n-tetradecane (C14).28 The system forms spherical PS domains within a PB/C14 matrix.

However, a temperature range was found below the dissolution point where the expected

higher order Bragg reflections were missing from the SAXS profile indicating the

absence of long-range order. At lower temperatures, evidence of higher order Bragg
reflections was found leading the authors to propose an ordering temperature, T,, below

which the spherical microdomains arrange themselves to form a cubic lattice. This

ordering process is illustrated in Figure 1-3.















---4c
CD


C-)


0
-<


m
U)


! MA


II


(I)

I-I

C)



CD
-H
II


The three domain morphologies commonly observed in diblock
copolymer systems.


0
C-<
F-
> :z
I
CO


U)
-D


m
U)


Figure 1-2


0

7
CD

0
--U
0
IF





0
-<




IF-
7>



0



I-
0
C)
r-H
o

rT

CO


























A DISORDERED ARRANGEMENT OF SPHERES


B / Cli

' 4
5 / cli


B / C14


B / 1
5 / Cli


AN ORDERED LATTICE OF SPHERES





Figure 1-3 An illustration of the ordering of spherical microdomains onto a cubic
lattice.








Microdomain ordering in block copolymers is an interesting analog to melting-

crystallization phenomena and as such is characterized by a nonconserved order

parameter (NCOP). Other systems such as ordering in "polyballs"29 have contributed

much to our understanding of these phenomena. The presence of this second type of

transformation in block copolymer solutions provides an additional opportunity to test

the applicability of present models of crystallization and behavior in NCOP systems.

An important feature of block copolymer materials is their great diversity in

industrial applications. These include their use as synthetic rubber, additives to plastics

and adhesives, and hydrophilic coatings.30 Much theoretical and experimental research

has been done in attempting to relate the mechanical properties of block copolymers to

their structural properties.31 32 For this reason, a better understanding of the structural

and thermal behavior of these materials is of direct industrial significance.

The SAXS technique is an excellent method for analyzing the structure of block

copolymers because the scattered intensity is directly related to the positional correlations

of the particles in the system. It was the goal of these studies to perform high resolution

x-ray scattering measurements of the equilibrium structure of block copolymer solutions

and to probe the microphase separation and ordering processes with fast kinetic

measurements. The successful completion of this experiment was dependent on three

major requirements. These were the use of high quality samples, a specially designed

polymer x-ray scattering furnace that allowed for rapid in-situ quenches, and a very

intense x-ray source enabling high resolution measurements. Ultimately, x-rays

produced by synchrotron radiation were utilized. The experimental methods employed

and the novel results obtained are discussed in the following chapters.

This dissertation is divided into seven chapters. Chapter 2 is a summary of recent

advances in our understanding of first order phase transitions and provides a background

for understanding the significance of these measurements. The third chapter is an in-

depth discussion of the general principles of x-ray diffraction and relates the relevant





8


physical parameters to experimentally measured quantities. Chapter 4 is a brief review

of the relevant literature on block copolymers. Chapter 5 is a detailed description of the

experimental apparatus and procedures performed in these studies. The

parameterization, analysis, and interpretation of the equilibrium results are presented in

Chapter 6 in preparation for discussion of the kinetic results in Chapter 7. Finally, the

important conclusions which are drawn from this work will be summarized in Chapter 8.













CHAPTER 2
PHASE TRANSITION KINETICS


This chapter is a brief review of the major developments in first-order phase

transition kinetics. The first section provides the definitions of many of the basic

concepts as well as an introduction to some of the general behavior. Broad comparisons

are made with higher order critical phenomena. The transformation process is divided

into three temporal regimes which are then discussed individually. These are the early,

growth, and coarsening regimes. Two distinct mechanisms are identified in the early

regime: nucleation and spinodal decomposition. During the growth regime,

transformation is characterized by growth laws and transformation curves. Finally, the

late stage of growth, coarsening, is described.


Phase Transition Kinetics General Concepts


Figure 2-1 is an example of the phase diagram for a binary alloy exhibiting a

phase transformation. The solid line on the plot represents the transition temperature as a

function of sample composition. The dashed curve is called the spinodal curve and will

be discussed shortly. The objective of kinetic studies is to observe the process by which

the transformation from one phase to another occurs. One method of probing these

phenomena is through the thermal quench. In a typical quench experiment, the system is
held at a temperature, Ti ( or other extensive variable ), above the transition temperature,

T, and then rapidly lowered to a fixed value below T,, Tf. If the quench occurs in a time

short relative to the time required for transformation, the system will still have the

















LU -_ T DISORDERED

COEXISTENCE
r0 [ CURVE
Ln SPINODAL
- -i- \- CURVE
LU
I __ UNSTABLE
SMETASTABLE

COMPOSITION








Figure 2-1 An example of a phase diagram for a binary alloy exhibiting a phase
transition.









structure of the initial phase. This phase is now unstable ( or metastable ) and will decay

to its equilibrium phase as time proceeds. The processes by which transformation occur

are often complex and the study of these processes is one of the most challenging topics

in non-equilibrium statistical physics.'
In order to quantify the transition process a quantity called the order parameter, 4,

is defined. This parameter takes on values between zero and one, being identically zero

in the disordered phase and one in the fully ordered phase. In some systems the order
parameter is a conserved quantity and in these cases, 4 changes only locally and remains

on average a constant for the entire system. Systems of this type are called conserved

order parameter ( COP) systems. An example of a COP system is a binary alloy which

exhibits phase separation.3 4, 5,6

A typical non-conserved order parameter ( NCOP ) system is a binary alloy

which exhibits an order-disorder transition.7, 8.9. 1.12.13 An example of such a system is

Cu3Au.7, 10 In the disordered state the crystal structure for this system is face-centered

cubic (FCC ) where the Cu and Au atoms are randomly distributed on all sites. In the

ordered state, the structure remains FCC with the Au atoms dominating the conventional

cube covers and the Cu atoms occupying the cube faces. The order parameter is related

to fraction of each species on a sublattice. Since the order parameter in this case is not

constrained by conservation laws, this is an example of an NCOP system. Other

examples of transitions characterized by NCOPs are melting/crystallization phenomena16

and antiferromagnetic ordering.14,15
The driving force for transformation is the difference in free energies between the

initial and final phases. Phase transitions may be classified as first, second, or higher

order based on a fundamental thermodynamic criterion according to the Ehrenfest

classification scheme.33 If the first derivative of the free energy of the system with

respect to the order parameter is singular, the transition is said to be first order. One

characteristic of first order phase transitions ( FOPTs ) which follows directly from the








discontinuity in the free energy is the existence of a latent heat (i. e. the heats of melting

and vaporization ). In addition transitions involving a change of symmetry from one

symmetry group to another that is not a subgroup of the first can be identified as FOPTs

because it is not possible for the system to go from one phase to the other continuously.

Discontinuity in the second and higher derivatives of the free energy with respect to the

order parameter without singularity in the lower derivatives implies second and higher

order transitions. Second and higher order ( also called critical or multicritical)

phenomena are in general much better understood than first order phenomena.1.2 This is

because critical phenomena arise from fluctuations in the order parameter which can be

characterized by a single correlation length. In fact, near the critical point the correlation

length diverges and only phenomena occurring on this length scale are relevant to the

transition behavior. It is the existence of this dominant length scale which has led to the

success of the renormalization group approach and the observation of scaling and self-

similarity in these systems.34 First order phase transitions are different because there is

in general no dominant length scale and transformation is determined by processes at the

interface between transformed and untransformed regions.

Mean field theories express the free energy, F, as a functional of the order

parameter, however, approximations are required in evaluating this expression. The

equilibrium state of the system can be determined by minimizing the free energy with

respect to <. The Ginzburg-Landau35 formalism describes an expansion of the coarse-

grained free energy in powers of the order parameter. In general, the order parameter

may be a scalar, vector or a tensor as is the orientational ordering of liquid crystals.2

The prototypical shape of the free energy curve for a scalar order parameter and
fixed composition is shown in Figure 2-2 for several temperature ranges. F{j) has the
double well structure characteristic of phase transformations. At temperatures above Te,

the lowest energy state corresponds to <=0. As the temperature is lowered, the difference

in free energies between the ordered, 4=1 and disordered, 4=0 configurations is reduced





























T > Tc



T Tc

Tc > T > Ts

Ts > T


The shape of the free energy curve versus order parameter for various
temperatures, above and below the transition point


Figure 2-2









until T=T(, at which point the free energy is equivalent for both phases. Below T, the

ordered phase is favored, but the state 0=0 is at a local minimum of free energy and is

therefore metastable. If the temperature is lowered further, the potential barrier

separating the ordered and disordered phases disappears and the <=0 state becomes

unstable. This instability limit which separates the metastable and unstable regions of the

phase diagram is commonly called the spinodal curve.'.2. 35 In fact it is no longer

believed that a sharp spinodal point exists2 as will be discussed momentarily. The

process by which the disordered state decays is qualitatively different for quenches into

the metastable and unstable regions of the phase diagram. In fact two distinct behaviors

are associated with these two regimes. When the initial phase is metastable,

transformation is said to result from the mechanism of nucleation. During nucleation,

localized droplets of the ordered phase spontaneously form within a disordered matrix.

These droplets grow independently until separate domains impinge on one another.

When the initial phase is unstable, it will decay through the mechanism of spinodal

decomposition. During spinodal decomposition long wavelength, infinitesimal

oscillations of the order parameter appear which are characterized by an amplitude and

wavelength that grow with time.

Many of the essential features observed in phase transition kinetics have been

elucidated through analysis of very simple models.' Perhaps the most useful of these is

the Ising Model because of the ease of calculation and the straightforward analogies to

physical systems.16 Within the Ising model, a lattice is composed of spins having values
of o = +- 1. By choosing the correct spin dynamics and an appropriate order parameter

this model may be applied to both NCOP and COP processes. For example, if changes in

the lattice distribution of spins occur by exchanging adjacent spins the system is said to

obey Kawasaki spin dynamics.' The total magnetization
M=Yoa (2-1)
i









remains constant throughout the transformation and is therefore analogous to the order

parameter in a COP system such as phase separation in binary alloys.36 In this case "up"

spins correspond to one type of atom while "down" spins correspond to the other type of

atom. The order-disorder transition in binary alloys can also be modeled using Kawasaki

dynamics if the order parameter is defined to be a sub=lattice magnetization:
M,= Doi (2-2)
sublattice

In this event, the total magnetization is conserved, but the order parameter is

nonconserved.37 38 Another way to model NCOP systems is with Glauber spin

dynamics. Within this model spins are able to change sign independent of adjacent spins

so that the order parameter which is taken to be the total magnetization is nonconserved.

In general spin dynamical models have shown good agreement with theory.16

Conceptually the transformation process can be divided into three temporal

regimes: early stage growth characterized by either nucleation or spinodal

decomposition, intermediate growth, and late stage growth commonly called coarsening.

These three stages are illustrated in Figure 2-3. During late stage growth, essentially all

of the system is in small fully transformed regions. During this regime the larger of

these domains will grow at the expense of the smaller domains resulting in an increase in

the average size of the domains. This process is termed coarsening and is illustrated in

frames c and d of Figure 2-3. The different time regimes and transformation

mechanisms will now be discussed separately beginning with nucleation and spinodal

decomposition, followed by a brief discussion of intermediate stage growth and finishing

with a description of coarsening behavior. The chapter will conclude with some

comments on recent efforts to develop a universal classification scheme.








Nucleation




Growth
0

Growth


0*


Early Coarsening


Late Coarsening


An illustration of the different temporal regimes in the transformation
process.


Figure 2-3








Nucleation Theory


The rate of transformation from the old to new phase is controlled by the

combination of the total nucleation rate and the growth rate. The total nucleation rate per

unit time, IT(t), is the product of the nucleation rate, I(t), and the volume available for
nucleation, V,. Conceptually, the behavior of the total nucleation rate can be divided

into three regimes. Often these three regimes are distinct but in some cases or there may

be significant overlap. In addition, not all of the regimes are necessarily observed. The

first regime is characterized by a transient nucleation rate.39 Following a quench from

above the transition point I(t) will be zero initially and then grow smoothly to its steady-

state value. Once the nucleation rate reaches its steady-state value, Is', it will remain

unchanged throughout the transformation. However, as the transformation proceeds the
nucleation volume, V,, decreases leading to a decrease in the total nucleation rate, IT(t),

in the late time regime. The process by which the nucleation volume is exhausted

depends in detail on the type of site in which nucleation occurs.40 Several of these

processes will be discussed in a later section. Each of these three regimes will be

discussed in detail in the following sections.

The driving force for transformation from the metastable state to the equilibrium

state is the free energy difference between the two phases.39 However, when a finite

sized nucleus forms, there exists an interfacial region between the nucleus and the bulk

metastable phase. The surface tension serves to raise the free energy of the droplet. The

total free energy of the embryo is typically written as the sum of bulk and surface terms:


AF, = n (f f) + So (2-3)


or AF,= [ (f f- f ) + 4xcr2 (2-4)










In the above expression, the droplet of radius, r, is composed of n particles. The product
of the surface area of the droplet, S,, and the surface tension, a, gives the surface

contribution to the free energy. The bulk free energies of the equilibrium and metastable

phases per particle are f" and f", respectively. The volume per particle in the

equilibrium phase is designated v". The volume free energy term f" is generally assumed
to be independent of droplet size. Similarly, a is assumed to be independent of size and

calculations of the surface tension exhibited by crystalline clusters suspended in liquid

support the validity of this assumption.41

The energies, fe and f", are in general a function of the reduced temperature,
t=(To-T)/To where To is the transition temperature. Above To, fe > fm and AFn is a

monotonically increasing function of n and fluctuations of any size tend to evaporate. In

this case, the equilibrium form of the distribution of sizes of nuclei as determined by

Boltzmann statistics is given by


f-AFn1
N. = N' exp (2-5)



which is only of significant magnitude at very low values of n. However, below To, f <

fn and AF, goes through a maximum at the critical radius, r,.


2ov*
rc =(ife) (2-6)



Classically, nuclei of this radius are in unstable equilibrium whereas nuclei with r < r,

will evaporate and droplets with r > r, will grow. This model, while being somewhat

simplified, exhibits the important feature that nuclei having radii near r, control the









nucleation rate. In other words, it is the rate at which fluctuations reach the critical size

that determines the rate of decay of the metastable state.

Immediately following a quench the majority of nuclei are of a very small size

and until the first droplets reach the critical size the nucleation rate is effectively zero.
This period is called the induction or incubation time, to.

There are two commonly used approximations for I(t) in the transient period.

The first assumes an incubation time followed by a slowly growing nucleation rate which

is approximated by a power law,


I(t) = ( t to ) B,t (2-7)


Here the step function, O(t), incorporates the effect of a finite incubation time. This

approximation will be most valid when the nucleation rate is a slowly changing function

of the time and the transformation nears completion before the nucleation rate

approaches its steady-state value. The second approximation incorporates an induction

time and a step function to describe I(t) and will be most accurate when, following the

induction period, I(t) grows rapidly to its steady-state value. Typically, the effect of the

changing nucleation rate is most obvious in the early stages of the transformation.

Experimental measurements are not overly sensitive to the exact form chosen to

approximate the time dependence of I(t) in this regime.38

In the second time regime the nucleation rate is constant at its steady-state value,

I". While there have been many attempts to calculate I" by far the most successful

treatment is that of Becker and Doring.42 The key to this model is its treatment of the

problem as a kinetic phenomenon. A set of difference equations is used to describe the
number of droplets as a function of size and time, Z,. The rate of change of the number

of droplets composed of n particles is expressed as the difference between the rate of

condensation and the rate of evaporation of particles from the nucleus. The condensation








rate is assumed to be proportional to the surface area of the nucleus, S., and otherwise

independent of n. As time proceeds, the number of embryos of size n approaches its
quasi-steady state value ( Z., = Z ). This implies that I, = I, a constant independent of

n so that the distribution of nuclei, Z7, is constant. There are two boundary conditions

for the kinetic process. The first is that for small n the distribution of nuclei approaches
its equilibrium form as determined by Boltzmann statistics (Z, approaches N, from

Equation 2-5 ). The other boundary condition is that embryos containing a very large
number of particles leave the system completely ( Z,= 0 n>>n,). By equating the rates

of production and loss of embryos of size n, the following result is obtained:



I-s = v (2-8)

n=l


where v is the rate of condensation per unit surface area of the nucleus. Since AF
exhibits a maximum at nc, the quantity N,-1 from Equation 2-5 will be sharply peaked

near nc. Becker and Doring reasoned that only values of n near the critical number
would contribute significantly to the above sum. Taking the continuum limit of N, and

AF, the standard Becker-Doring form for the steady-state nucleation rate is obtained:


I- = A exp (2-9)


fNvSc] AF(n.)
where A= 3J B = AF(n) (2-10)

In the above relations, both the prefactor A and the energy term, B, depend on the

reduced temperature. In those cases where these parameters are only weakly dependent

on the temperature, the steady-state value of the nucleation rate takes the Arhenneus35









form with an activation energy equal to the free energy difference between a critically

sized nucleus and the bulk metastable phase.


Spinodal Decomposition


The different modes of transformation observed for metastable and unstable

systems can be viewed as instabilities against different types of fluctuations as discussed

by many authors.1' 2.16 The method of decay of a metastable state is through the

nucleation process. This indicates an instability against localized, large amplitude

fluctuations of the order parameter, the size of which grows with time. This process is

illustrated in Figure 2-4. In contrast, the method of decay of an unstable system is

through a process termed spinodal decomposition. This process is also shown in Figure

2-4. This mode of decay indicates an instablility to nonlocalized or long wavelength

fluctuations of the order parameter of very small amplitude. The amplitude of these

fluctuations grows with time. While nucleation gives rise to growing droplets, spinodal

decomposition leads to complex interconnected patterns of the transformed phase.43 The

interfacial region between transformed and untransformed regions is initially much less

defined than that of nucleated droplets but sharpens as time proceeds. As the size of the

transformed domains increases, the patterns become less interconnected until the domains

are similar in shape to those observed for nucleated transformations.

The significance of the spinodal curve has recently been questioned by many

researchers2 17 because the concept of an instability limit is an artifact of mean field

theory used to formulate the free energy functional. However, real systems do show the

types of behavior described above. And although the transition from nucleated to

spinodal decomposition behavior occurs gradually26 instead of suddenly at the spinodal

point, the concept of a spinodal limit is not totally without merit.













NUCLEATION


ti \ t2 t t3

Distance





SPINODAL DECOMPOSITION


Distance


A contrast of the nucleation and spinodal decomposition transformation
mechanisms.


Figure 2-4








Domain Growth


For the types of systems we are considering, the growth of domains of the

equilibrium phase is said to be thermally activated. This implies that the growth velocity

is a function of temperature, being greatly reduced at temperatures low relative to the

transition temperature. One consequence of this behavior is the ability to "quench in" the

high temperature phase by rapidly lowering the system temperature to a value where the

velocity of growth is extremely small.
In addition to the dependence on temperature, the growth velocity, F, will also

have a characteristic dependence on the size of the domain ( or equivalently, on the time

elapsed since the birth of the domain). In general, thermally activated transitions can be

divided into two groups: interface controlled and diffusion controlled reactions.39 These

two groups are characterized by functionally distinct forms for the growth velocity.

Generally, transformations between two phases having approximately the same

particle concentrations are controlled by the energetic of processes occurring at the

interface. This being the case, the growth velocity, F, is independent of particle size and

time so that the average dimension of a growing droplet, L(t) satisfies the simple

equation.44,45
L(t) = ro t, F(t) = r (2-11)

Christian39 has estimated the dependence of F on the temperature, T, for the decay of

metastable phases. This calculation assumes the difference in free energy between the
old and new phases to be small relative to the thermal energy, kBT, where kB is the

Boltzmann constant. In addition, the characteristic frequency for particle motion, vo, is

assumed to be identical in both the transformed and untransformed regions. The

resulting relation is
(T) = ( AFv) e (2-12)
r(T) =(vo) kgT exp [- T(2-12)
^B1 I B1








Here 8 is the thickness of the interface and AH, is the activation energy separating the

equilibrium and metastable states which is generally a function of the reduced
temperature, T.

In the event that there is a significant concentration difference between the initial

and final phases, it is possible for the propagation of the domain boundary to be diffusion

limited. If the concentration of particles is higher in the transformed domain, there exists

a region adjacent to the interface called the depletion zone' in which the particle

concentration falls below its average value in the untransformed volume. In order for the

boundary to propagate, particles must diffuse through the depletion zone and then

"condense" on the transformed domain. If the processes occurring at the interface are

much slower than the diffusion of particles through the depleted region, the
transformation is interface controlled (i. e.. L(t) = F0 t). However, if the condensation

process is rapid, transformation will be limited to the rate of diffusion through the

depletion zone. From the diffusion equation and dimensional analysis Christian finds
D
L(t) ( Dt )z r(t) (2-13)

where D is the diffusion constant. This parabolic growth law is verified in more exact

calculations,46.47 and in computer simulations.48'49

It is noted that the above growth laws for interface and diffusion controlled

growth are only valid prior to impingement of one transformed domain on another.

Different growth laws govern behavior in the post-impingement or coarsening regime

and these will be discussed shortly.


Transformation Curves


Now that the processes of nucleation and growth have been discussed, a method

for combining the results to yield a rate of transformation will be presented. A measure









of the degree to which transformation has progressed is the fractional volume in the
transformed state, C(t). From the previous discussion of the nucleation rate we expect

this quantity to grow slowly at first and then more rapidly as a larger number of embryos

reach the critical size and begin to grow. Eventually, the surfaces of different domains
will impinge on each other, slowing the growth of ((t) at later times. The shape of a

typical transformation curve is shown in Figure 2-5. The relevant features include the
induction time, to, and the time to half-completion, ;,.

The earliest formal theories of transformation kinetics are due to Kolmogorov,50

Avrami,51-52.53 and Mehl, and Johnson.54 The argument of Avrami goes as follows. If a

nucleus forms at some time, t, then the transformed volume which is contained in the

nucleus is given by


V(t) =rlyxyY (t-t)3 (t>t) (2-14)
=0 (t
Here Yx, y,, and yz are the growth velocities in the x, y, and z directions respectively and
4L
Tr is a shape factor (for isotropic growth, rl = and y, = yy = y, = y). The above

expression is valid for a system exhibiting a linear growth law, L(t) t, such as is the

case for an interface controlled transformation as was previously discussed. In a system

where growth is diffusion limited, the growth law is given by L(t) tIA. In this case

Equations 2-14 would be replaced by


V = TlyYy z ( t -t)3' (t>r) (2-15)

=0 (t
For the remainder of this section only systems exhibiting a linear growth law will be

discussed with the understanding that the appropriate results for different growth laws

can be derived simply by making the modification shown in Equations 2-15.

































_ _
1/2 /


0-


TIME


A simulated transformation curve showing the induction time and the
half-completion time.


S-


C.


Figure 2-5


i 1 I


I








In early times there is little impingement of domains, so that each droplet grows

independently. In this limit the transformed volume is given by


V= 3-]f I(t) (t- T)3 d' (2-16)
0


where isotropic growth was assumed. For a constant nucleation rate, I(t) (I(t) reaches

the steady-state value, I", immediately ) Equation 2-16 is reduced to



= V 3 I3 t4 (2-17)



This confirms the rapid increase of C at early times.

Unimpeded growth will not continue for all times, however, due to impingement
by other droplets. To deal with this complication, a concept called the extended volume,
Ve is introduced. The extended volume is the volume of transformed material assuming

that nuclei continue to form not only in untransformed regions of the system, but in
previously transformed regions as well. Similarly, growth of domains is assumed not to

stop when impingement occurs; rather, each domain continues to grow through the
others. In this view, Ve is equivalent to VB in Equation 2-17 for all times. In any

increment of time, dt, the probability that the incremental increase in the extended
volume occurs in a previously untransformed region is ( 1 VP/V ). It follows that


dVP= (1 VB/V) dVe (2-18)

or = vP/V = 1 exp( -Ve) = 1 exp(-T_ I" t4)








The shape of this curve is shown in Figure 2-5 and in the early time this expression

reduces to Equation 2-17.

In the above argument, the nucleation rate I(t) was assumed to be a constant. In

general, the total nucleation rate is a function of the time, either due to transient

nucleation in the early regime or due to nucleation site saturation in the later regime.
The corresponding transformation laws may be derived in a straight forward way by

inserting the appropriate expression for I(t) into Equation 2-16. These calculations

generally lead to an expression similar to that in Equation 2-18 except that the time

exponent is different from 4. This led Mehl, Johnson, and Avrami ( MJA) to propose

the following general transformation law39:


= 1- exp( -ktn) (2-19)

For a constant nucleation rate it has been shown that n=4. For an increasing nucleation

rate ( early regime ) n>4, and for a decreasing nucleation rate ( site saturation) n<4.

For homogeneous nucleation, often the nucleation rate may be approximated by a
step function being zero during the induction period, to, and then taking on its steady-

state value Is. In this case, the transformation curve is given by Equation 2-19 with n=4
and the time, t, is replaced by t-to.

Often nucleation occurs not homogeneously, but at impurity or defect sites within

the sample. The transformation curves for heterogeneous nucleation can be calculated in

much the same way as for the homogeneous case if the appropriate form for the
nucleation rate can be found. For instance, when nucleation occurs spontaneously at No

impurity sites the effective nucleation rate, I,(t), may be represented by a delta function at

t=O having a strength, No; i. e..


I(t)= No 8(t)


(2-20a)








so that from Equation 2-17,


= 1 exp( N 'y t3) (2-20b)


The form of the transformation curve is unchanged, with n=3 reflecting the nature of the

nucleation process.

Interesting dimensional effects result if nucleation is spontaneous and growth is

restricted to either a lamellar or rod-like volume. For growth occurring in a sheet ( 2-D

growth ), as might be the case in a thin film, the extended volume is given by :


VP = (47cV6)) No y2 ( t x )2 (2-21)



where 8 is the thickness of the sheet. The result for a constant nucleation rate is the MJA

expression with n=2. Similarly, it is seen that for growth in a rod-like volume ( 1-D ) the

result is MJA with n=l.

More complicated transformation curves may result if nucleation occurs

heterogeneously, but not spontaneously. If defects are located randomly, the growth in

early times should be indistinguishable from the homogeneous case ( Equation 2-19 with

n=4). In time the available nucleation sites may be exhausted before transformation of

the entire volume is complete. In this event, different behavior is expected following

saturation of the available sites. Cahn derived an expression for nucleation occurring on

grain boundaries, edge dislocations and point defects.40 Analogous to the concept of an

extended volume for transformation, an extended volume for nucleation is introduced. In

this way, the space available for nucleation at time, t, is calculated as a function of the

growth velocity, steady-state nucleation rate, and the initial volume available for
nucleation, Vn. The resulting transformation curve for nucleation occurring on two

dimensional defects is :










S= 1- exp(-b f(t)) (2-22)
1
where f(t) = ( 1 exp(-' ( (1-32+2l3)]d
0


The coefficients a and b are related to the physical quantities y, I", and V, in the

following way:


a= ( Iss )1t3 (2-23)

and b = 2Vn l



The function f(t) has two limiting forms, at late times f(t) t, and at early times f(t) t
as expected. The transition from n=4 before site saturation to n=l after saturation is best
illustrated by a plot of In( ln( 1/(1-) ) ) vs In( t) as shown in Fig. 2-6. In this plot the

slope of the curve is equivalent to the exponent in the MJA form, and the bend in the line
corresponds to site saturation. From Equations 2-21 it is evident that if we offset the plot
vertically by In( b ) and horizontally by ln( a) we have a master curve for this type of
transformation. In other words, plots for a number of data sets will fall upon this master
curve if offset by the appropriate selection of a and b on this scale.
Similarly, the expressions for the transformation curves for nucleation on line and
point defects have been derived. In both cases the early time behavior agrees with the
MJA form having n=4. In late times, n crosses over to 2 for nucleation on line, and 3
for point defects. It is apparent that the spontaneous nucleation of point defects discussed
previously is just a special case of this result where only the n=3 regime of the



















The Cahn Model


8

S6


4


2

0


-2


C
-4


-6
- -6 -


-8
-1


Master Curve


0 1 2 3 4


In (


Figure 2-6


time )


An ideal Master Curve for a transformation which obeys Cahn's
relation for nucleation on two dimensional nucleation sites.


n = 1


n = 4









transformation curve is observable. In fact, the easily observable range of C in

experiment is typically from about 0.01 to 0.99. Although it seems that this range should

encompass the full range of kinetic behavior, this range corresponds to only a relatively

small portion of the master curve. If the nucleation rate is too high, saturation occurs

very early in the transformation and only the behavior of saturated regime is evident.

Conversely, if the nucleation rate is too low, saturation occurs late in the reaction and

only the exponent corresponding to the unsaturated or homogeneous regime is seen. In
fact, the crossover is only visible if saturation occurs when t 0.5. The steady-state

nucleation rate within the Cahn theory for nucleation on 2-D defects which corresponds

to this point is given by :

r i3
I-s = V3 (2-24)




Systems exhibiting the required balance between the nucleation rate, growth velocity,

and nucleation volume appear to be exceedingly rare. As will be discussed, the

observation of this phenomenon is one of the more fascinating features in the study of

ordering in block copolymers.


Late Stage Growth Coarsening


Often the equilibrium state toward which the system is evolving is degenerate.
For example, in the Cu3Au system described previously there are four identical sites in

the unit cell. The lattice can be defined by choosing any one of these four sites as the

corer site. In this way it is seen that the ground state degeneracy of the ordered phase is

p=4. For the crystallization process, p=o reflecting the fact that the crystal may form in

any orientation within the liquid.39








In the event the ground state is degenerate, in the late time regime the system will

be composed of many fully transformed domains distributed in the p degenerate states.

As time proceeds, the number of domains will decrease with a corresponding increase in

the average size of the ordered domains. This process is called coarsening.

Within the coarsening regime, the system is self-similar with respect to scaling by

length and time. This can be illustrated by zooming in on frame (c) of Figure 2-3 and

comparing the result with frame (d). The relevant features in each case will on average

be identical. A well known consequence of this type of scaling is the existence of a

characteristic length37, L(t), (the domain size) where

L(t) t (2-25)

Here t is the time following the quench. The exponent a has been seen to be different in

COP and NCOP systems. Specifically, theoretical calculations55 and experimental

observations56 of COP systems support the assertion a=l/3. In NCOP systems the

growth rate of domains in the coarsening stage is determined by energetic at the

interface and is often curvature driven. Theoretical calculations predict a corresponding

growth exponent, a=l/2.57 This result is supported by a growing number of experimental

findings7.58 and computer simulations59,60. It should be noted that the growth exponents

are different in the coarsening regime from those discussed for the growth regime. This

is because during the early stages, growth of domains is relatively uninhibited by

impingement. In the coarsening regime, the domains are strongly interacting with the

major mode of growth being the absorption of smaller domains by larger ones. Also, the

dynamic quantity during coarsening is the average size of a domain, while the volume

fraction of the system in the transformed phase remains approximately unchanged.

In the early stages of transformation it was seen that impurities and defects have a

profound effect on the nucleation process by providing heterogeneous nucleation sites.

These entities can also lead to interesting effects in the coarsening regime. Lai et al.61








show that random fields which are generated by impurities and defects slow the

coarsening process so that the growth law becomes:

L(t) (log(t) )m (2-26)

where m depends on the detailed nature of the imperfections in the system. Evidence
supporting this assertion has been provided by experiments on Cu3+xAu where excess Cu

atoms appear to behave as diffusive impurities.10


Universal Classification


The non-equilibrium scaling behavior observed at late times is reminiscent of the

scaling seen in critical phenomena. For this reason, one of the foremost efforts in the

study of phase transition kinetics has been the attempt at developing a universal

classification scheme analogous to that for critical phenomena.17 In this way it is hoped

that the relevant parameters such as ground state degeneracy, dimensionality, and

conservation property of the order parameter can be identified so that the behavior of

widely diverse systems can be better understood. The coarsening growth law is one

feature which may be useful in identifying these universality classes. For systems

exhibiting random fields, the growth law is logarithmic suggesting that systems with

different types of impurities might form separate universality classes. The growth

exponent a has been seen to depend on the type of order parameter. It is, however,

independent of ground state degeneracy, and dimensionality in many cases.

In order to better distinguish the relevant parameters for classification and

provide a deeper understanding of the fundamental processes which govern phase

transformation, a broad range of physical systems must be investigated. Polymers and

block copolymers in particular are excellent systems for this purpose.1 An excellent

method for observing the transformation process in these materials is through the small





35


angle x-ray scattering spectrum. The fundamentals of the x-ray scattering technique are

now presented.











CHAPTER 3
THE X-RAY SCATTERING TECHNIQUE




The x-ray scattering technique is an excellent probe of inter- and intraparticle

correlations. Many excellent reviews of general scattering theory can be found in the
literature.62.63.64,65.66 A brief discussion of the fundamentals of x-ray diffraction is given

here followed by some details of intraparticle effects in small angle scattering.

Interparticle interference effects resulting from a system of particles with liquid-like

correlations are then described. Finally a description of the scattering from an ordered

array is presented including the effects of grain size, strain, and temperature.


Fundamentals of X-ray Diffraction


When a beam of x-rays is incident on a sample, a characteristic scattering pattern

results. In the systems with which we are dealing, the diffraction of x-rays is due almost

entirely to elastic scattering from the electrons in the material. The scattering

distribution from a single electron is given by the Thomson formula:67
Ie(20) = I0 m2 (3-1)

The angle 20 is called the scattering angle and is illustrated in Figure 3-1. Here OP is a

polarization factor which equals 1 for an incident beam polarized perpendicular to the

plane of scattering ( as is the case for synchrotron radiation). For an unpolarized beam,
l+cos2(20)
= 2 (3-2)








In either case the polarization factor is approximately equal to unity in small angles.
Compton and inelastic scattering are negligibly small at small angles (i.e. 20 < 20 ).64

The elastically scattered waves are coherent, meaning that the total scattered

amplitude is the linear superposition of the electric field vectors for each scattered wave.

The scattering from two scattering centers is illustrated in Figure 3-1. If the amplitudes

of the two scattered waves are equal and of unit magnitude, the total scattering amplitude
will be determined by the relative phases of the two waves. The phase for each wave at
the point of detection equals 2nx/ times the optical path for each of the two waves.

Actually, there is an additional factor of n which results from the phase inversion

incurred by scattering from an electron. Since all scattered waves are offset by this

factor, the total phase shift is given by
27t
= r.- (- so )=-r.(k-ko ) (3-3)

Here, so and s are unit vectors in the incident and scattered directions, and k k are wave

vectors in the incident and scattered directions. At this point it is convenient to define
the scattering vector, Q, which is the difference in incoming and outgoing wave vectors

so that
[20 4E f29
Q = k-ko, Q= 2k sin = sinm (3-4)

The scattering vector is the variable typically used in x-ray diffraction and henceforth the

scattering will be described in terms of this quantity. This is done with the understanding
that Q can be related to the scattering angle, 20, by Equations 3-4. The transformation

from Q to scattering angle is approximately linear at small Q with 10 = 0.07121 A-1 for

an incident wavelength of 1.541 A.
With these definitions, the amplitude of scattering from two scattering centers is
A = 1 + exp(-i4) = 1 + exp( -iQ-r) (3-5)

































So















An illustration of diffraction by two scattering centers and the definition
of the scattering angle, 20.


Figure 3-1


7-(s-So)








This result can be extended to the case of scattering from a continuous system
characterized by an electron density, p(r). The resulting amplitude is given by

A(Q) = fdVp(r) exp( -iQ-r) (3-6)


Hence, the scattering amplitude is equivalent to the Fourier transformation of the electron

density. The observed scattered intensity is related to the amplitude in the following

way:


I(Q) c AA*= f dV1 dVz p(r1) p(rz) exp(-iQ-(rl-r2)) (3-7)



Redefining r = r, r2, in Equation 3-7 yields the result that the scattered intensity is

equal to the Fourier transform of the auto-correlation function of the electron density,
-2
p (r):

I(Q) f dV 2(r) exp( -i Q-r) (3-8)


where p (r) = dV, p(r) p(r2)


This is a very general result which relates the positional correlations of electron density

fluctuations in the system to the measured scattering profile in the absence of

approximations. In cases where a discrete treatment is more appropriate, a particle-
2
particle correlation function replaces the term, p (r), in Equation 3-8.68

Often the systems studied with scattering methods are composed of particles
having an approximately uniform electron density, p,. These particles are suspended in a

gel or solution characterized by another electron density, P2. Neglecting inhomogeneities

in the charge density, the scattering is equivalent to that from a system of particles with
density Ap = p, P2 superimposed on a background of density P2. There will be no








observable scattering from the background in the relevant scattering range. The problem

is therefore reduced to the much simpler case of calculating the scattering from particles
of density Ap suspended in a vacuum.64

The scattering amplitude for a system of N identical particles can be written
N
A(Q) = fp(r) exp( -iQ-( R + r )) dV (3-9)

k=1
N
= F(Q) exp(-iQ.Rk)
k=1
F(Q) = fp(r) exp(-iQ-r) dV

Here Rk is a vector extending from the origin to the center of symmetry of the kth

particle and p(r) is the electron density distribution of a single particle about its center of

symmetry. The quantity, F(Q), called the form factor for the particle, contains all

information regarding intraparticle correlations. The structure factor, S(Q), is the

product of the scattered amplitude, A, with its complex conjugate and is a weighted sum

of the phase factors corresponding to the center of mass positions of all of the particles.

S(Q) therefore contains information concerning interparticle correlations. The scattered
intensity is proportional to S(Q) which can be written :
I(Q) c S(Q) = F(Q) F*(Q) cos( Q.( R, R) ) (3-10)

k
After averaging over all particle orientations and separating those terms where

j=k, this expression ( Equation 3-10) for S(Q) becomes
S(Q) = N< IF(Q)2 > + NI< F(Q) >12 C cos( Q-(Rk-RJ)) (3-11)
kj
The pair correlation function, P(r), introduced by Zernicke and Prins69 represents the

probability that two particles are separated by a distance r. In the case of an isotropic

distribution of particle positions and orientations, this can be used to evaluate the double

summation in Equation 3-11. The resulting form for S(Q) is








00
< F(Q) >2 sin(Qr)
S(Q) = N < F2(Q) > 0
where v, is the average volume available per particle in the system. As v, decreases,

interparticle interference effects become more pronounced. With v, large S(Q) is

dominated by the first term in Equation 3-12. In this case, the observable scattering is

just that of a single particle multiplied by the number of particles in the system. This is
the limit typically explored in SAXS and the detailed properties of the scattered intensity

in this regime will now be discussed.


Small Angle X-ray Scattering


For particles with dimensions on the order of hundreds of angstroms and with a

typical incident x-ray wavelength of 1.5 A, the majority of scattering features appear in
the range from 0 to 2 degrees. This is typical of block copolymer solutions. Assuming a
dilute, isotropic system of particles, the scattered intensity can be calculated from

Equations 3-8. In all but the simplest cases this must be done numerically. For the

important case of a hard sphere, the structure factor, first derived by Rayleigh,70 is
S= Ap )2 V 3 sin(QR) QR cos( QR)2
ere (Q12 ( )2 V2 (QR) (3-13)

where V is the volume and R the radius of the sphere. A plot of this function for three

radii is shown on a linear and on a log scales in Figure 3-2. For the larger radius, the

scattering function decays more rapidly reflecting the inverse relation between length

scale and scattering vector. This function exhibits well defined maxima at the locations

QR = 5.77, 9.10, 12.32, etc. from which the sphere radius can be determined.

A more general method for determining the average size of particles exhibiting
small anisotropy was developed by Guinier.71 In the limit of very small angles, Equation











0
4-j
0
0
L-

E
0

0
L-
O



0












O
LL)
0)







0
U-






Ou
t-
LL
O
U


L
UL)


10


1 -1


10-2


10-3
10


10-5 L
10.00
0.00


1.00


0.80


0.60


0.40


0.20


0.00 -
0.00


0.02 0.04 0.06 0.08


Scattering Vector


0.13


A-1
A


Figure 3-2 The scattering form factors for and ideal spheres having radii: 80, 100,
and 120 A.


3.02 0.04 0.06 0.08

Scattering Vector -1
Scattering Vector ( A )








3-8 reduces to
-2<
S(Q) = ( Ap )2 Vexp (3-14)

fp(r) r2 dV
where Rg = the radius of gyration = M

Here the integral is taken over a single particle and M is the total mass of the

particle. Equation 3-14 shows that if the log of the scattered intensity is plotted versus

the square of the scattering vector ( called a Guinier plot), the slope of the resulting line

will be proportional to the square of the radius of gyration in the limit of very small

angles. Thus if the shape of the particle is known, the parameters describing its size may

be determined from the radius of gyration as obtained from the scattering profile. For
example, the radius of a sphere is related to its radius of gyration by Rg = [3/5 R. The

Guinier approximation breaks down for severely anisotropic particles (i. e. plates

or rods) and is invalid in the presence of interparticle interference effects.

In practice, all of the particles do not have identical radii. Rather, there is a

distribution of radii centered about a mean. In the case of block copolymers this effect

arises from the fact that the polymer chains themselves are polydisperse. For example

the scattering which results from a system of spheres characterized by a gaussian size
distribution, having a mean radius of R0 = 100 A, and a standard deviations, a = 1, 10,

and 20A are plotted in Figure 3-3. The number of particles, N, having radius, R, is

given by

N(R)= exp R-)2 (3-15)

and the total structure factor is the superposition of the structure factors from each

particle in the distribution,
S(Q) = fN(R) S pbe(Q,R) dR (3-16)











1.00

0.80

0.60

0.40

0.20


L-
0
4-a
0
0
LL
E
L-
0
LL

0
0

-c
L_
/)


0
I,
0
LL

E
0
L-


0


CL
c--


100


10


10-2


10-3
10


10-4 1
0.00


0.02 0.04 0.06 0.08 0.10

Scattering Vector ( A )


0.02 0.04 0.06 0.08 0.10


Scattering Vector ( A ')
A plot showing the effect of polydispersity on the scattering form factor
for a gaussian distribution of spherical particles having a width parameter,
a = 1, 10, and 20 A.


0.00 1
0.00


Figure 3-3








The major effect of polydispersity is a weakening of the scattering minima which is more
pronounced with larger o.

Typically, the interface between the particles and background is of finite width.
For block copolymers this region is where the two covalently joined blocks meet. If
these junction points form a perfect surface, the interface is ideal. It is often assumed
that the locations of these junction points are described by a gaussian distribution of
width, This leads to a density profile
p(r) = Ap 1 erf[ (3-17)

The structure factor calculated from Equations 3-6 and 3-17, incorporating the effect of
the interfacial thickness, 4, are shown in Figure 3-4. The spheres in this calculation have
a mean radius of 100 A and values of = 1, 10, and 20 A. There is relatively little

perturbation of the profile at very low Q, but the scattering at higher Q is greatly
suppressed for large values of 4. In addition slight shifts in the locations of the scattering
minima are observed.
At large values of the scattering vector, a different limit is found. In the high Q
region, the expression for the structure factor takes on a much simplified form
27n( Ap )2
S(Q) S (3-18)

In this equation, Sa is the surface area of the scattering particle. This power law

dependence is called Porod's Law.72 The radius of gyration and the surface area of the
particles can be determined from the limiting forms given by Eqs. 3-14 and 3-18 and are
very useful in deducing the shape and dimensions of the particle. If the surface of the
particle is of finite thickness the scattering at high Q will exhibit systematic deviations
from Porod's Law which depend on the interfacial thickness 4.
In a method developed by Vonk73 and Ruland,74 the effect of finite 4 is taken into
account by considering the electron density of the particle. p(r), as the convolution of an
ideal electron density profile, p,(r), and a smoothing function, h(r). Since the scattering











0

0


E
L-
0
O


0
.-
O





L..
-4




U
0
i-

E
L-
Q.
0
U-






U)


1.00
-

0.80

0.60

0.40

0.20

0.00
0.00



10


10 -
10.

ic2
10-2


o-3


-4
10 -

10-4
0.00


Scattering


Vector


The modifications to the scattered profile which result from spheres
having imperfect boundaries are illustrated in these plots for a system of
spheres having a radius of 100 A and values of x = 1, 10, and 20 A as
described in the text.


0.02 0.04- 0.06 0.08 0.10
Scattering Vector ( A
Scattering Vector ( A )


0.02 0.04 0.06 0.08 0.10


Figure 3-4








form factor, F(Q), is the Fourier transform of the electron density, p(r), it can be

expressed as the product of the form factor for the ideal particle and the Fourier
transform of h(r), Fh(Q). As described previously, h(r) is often approximated by a

Gaussian having a width parameter equal to In this case the total structure factor takes

the form
S(Q) C Q4 exp( -4 X2 2 Q2) (3-19)

Consequently, a plot of Q4 I(Q) vs Q2 yields the parameter, Application of this

method to the study of block copolymers is discussed in Chapter 4.
The above discussion was based on the assumption of widely spaced scattering

centers so that the total scattering is the sum of the individual contributions. Now, the

effect of a close packing of particles and interparticle interference will be discussed.


Scattering From Liquids


In the event that interparticle interference is present the scattering is given by

Equation 3-12. In general, the pair correlation function, P(r), depends on the average
volume per particle, v1, the temperature, T, and the interaction potential, Q(r). c(r) is

the two particle potential, i. e. the potential energy of a system of two particles separated

by a distance r. The determination of P(r) from the interaction potential is an extremely
difficult problem which has been extensively studied.7577 The approach developed by
Born and Green76 assumes a modified Maxwell-Boltzman dependence of P(r) on Q(r).

Their final result for the liquid structure factor is

S(Q) = N < F(Q)2 > + < F(Q) >2 (2-)3l(Q) 1- (3-20)

where the function 3(Q) is defined as



0








Here e is a constant of the system which is on the order of unity.
As an illustration of the above form, Equation 3-25 is evaluated for the hard
sphere potential :
I(r) = 0 r =0 r>Ro

The resulting profiles for four values of where vo is the particle volume are plotted in

Figure 3-5. Interference effects show up in the form of a reduction in intensity at very
low Q, and an enhancement at intermediate Q (Q = 0.03 A-, ). The result is a broad
maximum which grows with increasing concentration. Scattering at higher Q ( Q >
0.045 A-' ) remains mostly unaffected with the exception of a subtle shift in the position
of the primary spherical scattering maximum (Q = 0.06 A- ).
The Van der Waals force is the origin for the interaction commonly occurring in
many systems. The potential describing this interaction is well approximated by the
Lennard-Jones ( L-J) form :

(](r) = (o 12 (3-23)

This potential along with Equations 3-20 and 3-21 was successfully used by Fournet78'79
to predict the scattering from liquid and gaseous argon. Qualitatively, the scattering
curves are very similar to those of Figure 3-5. One distinguishing feature is a sharp
upturn in the scattering at small angles resulting from the long range of the L-J potential.
In addition, a secondary interparticle interference maximum appears on the high Q side
of the primary interference maximum.
From the above discussion, it is apparent that the scattering at high Q in a
concentrated system shows little deviation from single particle scattering. At lower Q,
strong oscillations about the single particle scattering become evident and take the shape
of broad peaks in the spectrum. In the case where the free energy favors a crystalline






49








The Hard Sphere


101


0
O
0

U-

0



CO

O"

-J













Figure 3-5


0
10
B

C
D
-ff


10
E



10-2





0.00
0.00


0.02 0.04 0.06 0.08

Sctterin Vec-1or ( A
Scattering Vector ( A )


".10


Model scattering profiles are shown for systems of spheres interacting
via the hard core potential at varying particle concentrations, vl/vo.


Model









arrangement of particles, the assumption of an isotropic distribution of particles is no

longer valid. The scattering from ordered structures will now be discussed.


Scattering From Ordered Structures


A crystal is a three dimensional periodic structure.0 The repeat unit for the

structure is called the unit cell. The lattice is constructed by translating the unit cell by
linear combinations of the basis vectors ( designated a,, a2, a3 ). A structure constructed

in this way can be considered to be composed of planes of particles as is shown in Figure

3-6. The scattering amplitude from such a structure will be determined by the phase

difference between waves diffracted from adjacent planes. If the path difference is equal

to an integer number of wavelengths, there will be constructive interference otherwise

the scattering amplitude will be zero. This is a qualitative statement of Bragg's Law62

which in its most common form is

mX = 2d sin( 20/2) (3-24)

Here d designates the distance between adjacent planes, X the wavelength of the scattered

waves, and m the order of scattering. In general, adjacent planes are separated by
al a2 a3
h k and in the three crystallographic directions. The integers (h,k,l) are called the

Miller indices and are used to specify crystallographic planes.

Calculating the distance between and the orientation of every group of
crystallographic planes is a formidable task. The concept of a reciprocal lattice is useful

in dealing with this type of calculation.80 In this theoretical construction three reciprocal

lattice vectors are defined


2( a2 x a3 ) 2x( a3 x a ) 27( aI x a2 )
a,* a2xa3 a *a2 xa3' a a, axa3



















































An illustration of Bragg scattering from a crystal lattice.


Figure 3-6








From vector algebra, one is able to show that the Bragg condition is equivalent to the
requirement that the scattering vector, Q, be a linear combination of the reciprocal lattice
vectors:
S= hb1 + kb2 + lb3 (3-26)

Within this construction, Bragg peaks are easily identified and designated by the Miller
indices h, k, and 1. For Bragg's law to be satisfied, the angle of incidence relative to the
set of scattering planes must equal the angle of reflection. This implies that Q is required
to be perpendicular to these planes and, for an orthorhombic system, has magnitude
Q = = 2r h + +2 1 )2 (3-27)


Thus, constructive interference from a perfect crystal will only occur at specific values of
the scattering angle, 28, and for specific orientations of the crystal relative to the

incident beam.


The Effect of Finite Size


A real crystal is generally composed of a large number of crystallites of finite
size. The scattered intensity from a single finite-sized crystallite having n particles in the
unit cell, and N,, N2, and N3 unit cells in the x, y, and z directions can be calculated62

and is proportional to the structure factor S(Q),


S(Q)= A A* sin2( N Q'al / 2) sin2( N2 Qa / 2) sin2( N3 Qa / 2)
sin2( QaI /2) sin2( Q-a / 2) sin2( Qa3 / 2)



where A(Q) = F(Q) eiQri









In the above expression the scattering amplitude, A(Q) is a function of the form factors,

Fi(Q), for n particles at positions ri within the unit cell. This function is peaked when the

following three conditions are satisfied:
Q-ai = 2xh, Q-a2 = 2xk, Q-a3 = 2rl (3-29)

or equivalently, when Q is a linear combination of the reciprocal lattice vectors.

In deriving Eq. 3-28 it was assumed that the grains were perfect parallelpipedons
of uniform size. In the limit of large N1, N2, and N3 and in real systems having a

distribution of grain sizes and shapes, the scattering lineshape is typically well
approximated by the gaussian, having an amplitude, A and a width, af.



The Crystal Structure Factor


From the above discussion it would appear that there exists a peak in intensity

corresponding to all integer values of h,k, and 1. This is true for a simple cubic structure.

However, for any structure having more than one particle in the unit cell, the structure
factor vanishes for certain combinations of h,k, and 1. The vector, r, from Equation 3-33

which specifies the location of each particle within the unit cell can be written as
ri= x, a + y, a2 + zi a3 (3-30)

where xi, Yi, and zi are belong to the interval: [ 0, 1]

The scattering amplitude can now be rewritten as


A(Q) = 2Fi(Q) ei( hxi +y + ) (3-31)
i=1

For a body-centered cubic structure ( BCC), there are two particles in the unit
cell having coordinates : x, = y, = z, = 0, and x2 = y2 = zz = 1/2. The scattering

amplitude for identical particles is








A(Q) = 0, h+k+1 odd (3-32)

A(Q) = 2 F(Q) h+k+1 even
So the crystal structure factor, SBCC(Q) = A A*, vanishes at those peak locations where

h+k+l is odd, and at those locations where h+k+1 is even, the peak is present having an

effective form factor equal to twice that of the constituent particles.

In a similar way, the scattering amplitude can be determined for the face-centered

cubic (FCC) structure,
A(Q) = 0 h, k, 1 mixed (3-33)

A(Q) = 4 F(Q) h, k, 1 unmixed
Only those peaks denoted by indices that are all even or all odd will be present ( i. e.

(111), (200), (311) etc.).


The Effect of Particle Motion


Until now it has been assumed that the particles are stationary on their lattice

positions. In fact, the particles exhibit displacement oscillations about their equilibrium

lattice positions. In conventional crystals, these motions are the thermal vibrations of the

atoms. For the block copolymer system, thermal oscillations for the massive

polystyrene domains suspended within the viscous polybutadiene matrix are negligible.

In these materials, the displacements result from diffusive motion of the domains in the

vicinity of their lattice positions. The change in the diffraction profile due to particle

motion can be described as a modification of the form factor in those cases where the
time characterizing particle motion is much shorter than observation times.
At any time the position of the kth particle can be written Rky = Rk + B(t). In

x-ray scattering we measure the temporal average (indicated by triangular brackets ) of

the scattered intensity, which is proportional to the structure factor:









S(Q) iQ-(Rk-RI) iQ-(S(t)-(8(t))
S(Q)= Fk2(Q) e < e > (3-34)

k
If uk(t) and u,(t) are defined to be the amplitudes of oscillation in the direction parallel to
Q, then equation 3-34 can be rewritten as
S(Q) = S(Q) + Sf(Q) (3-35)

2 iQ-(Rk-RI)
SI(Q) = > ^F,k(Q)e- J IFi(Q)e Je

k
The second contribution in Equation 3-35, Sd, contains the cross term, < ukUl > which is

vanishingly small unless the kth and Ith particles are very close. The result is a scattering
component that is slowly varying with Q and of very low amplitude. This contribution
to the scattering is called the thermal diffuse scattering and is approximately constant in
the high Q region of the spectrum.
The first term, SI, is identical to the result for the stationary finite-sized crystal

with the form factor replaced by an effective form factor
Feff(Q) = F(Q) exp( -- Q2< u2>) (3-36)

The exponential in the above equation is called the Debye-Waller factor. The peak
widths and positions are unchanged by displacement oscillations, but the peak amplitudes
are modulated by this factor.


The Effect of Strain


In a typical experiment scattering occurs not from one, but from many
crystallites. In general there exist random strain fields within the sample. The result is a
collection of grains having a narrow distribution of lattice constants centered about the








equilibrium value. If we define the strain, e = Aa/a, then the distribution of strains may

be approximated by
1 e2
P(e) ---exp( ) (3-37)
2 OY 20,

For simplicity we have assumed an isotropic gaussian distribution of strains and a cubic
lattice. The magnitude of the scattering vector at the bragg peak position is related to the
lattice constant via Eq. 3-27. The Bragg peak will be shifted by, AQ, if the lattice
constant is stretched by Aa. The peak shift and strain are related in the following way:

AQ = Q (/a) Aa = Q e (3-38)

so that the distribution of peak offsets is given by
1 (AQ)2
P(AQ)= exp(- 2 ) OQ = Q (3-39)
V2itOQ 20Q

The scattered intensity is the convolution of this function with the profile for a strain-free
system. The scattering profile for a strained system in this approximation is the
convolution of two gaussians which is itself a gaussian having a width equal to the sum
of the individual widths added in quadrature:
(Q-Qo)2 2
I(Q) = A (f/o) exp(- 2a ), O = ( (o+ (a.Q)2 )1 (3-40)


So the effect of strain is to broaden the peak by aoQ with an accompanying decrease in

the Bragg amplitude so that the integrated intensity of the peak is conserved. It is evident
that strain broadening is much more pronounced in the higher order peaks.


The Powder Pattern


If the orientation of the crystallites is random, the sample is called a powder. The
scattering will then be a function of the magnitude of Q only. Assuming a Gaussian








lineshape for the non-powder averaged Bragg reflection, the powder averaged structure
factor can be calculated


(Q) = (Q-Q0)21
Q (27 2)3/2 < exp- 22 > (3-41)

Here the brackets indicate an average of the structure factor over all orientations of the
Bragg center, Q0, relative to the scattering vector, Q. The form factor which has been

incorporated into A(Q) is assumed to be isotropic. Without loss of generality, Q is
chosen to lie in the z direction:


Q = Q Z, Qo = Qosin(0)cos(()) + Qosin(0)sin( )) + Q0sin(0)2 (3-42)


Here 0 and 4 are the conventional spherical coordinates. The average over 4 and 0
yields


Ao (Q-Qo)2 (Q+Q)2
S(Q)= Q [exp -Q +exp [- ) (3-43)



The second exponential is negligible for all positive values of Q.
There is an additional factor called the multiplicity, mnh which arises from the

fact that Bragg peaks having different indices h, k, 1 appear in the same location in the
powder spectrum. The intensity is proportional to the degeneracy of the Bragg
reflection. As examples, the multiplicities of several peaks for a simple cubic lattice are
listed below
(hkl)= (100) m= 6 (3-44)
(110) m= 12
(321) m= 48








It is possible to identify different crystal structures by comparing the relative amplitudes
of the observed Bragg peaks due to differences in the multiplicities of the different
structures. This will be discussed in greater detail in Chapter 6.
If the effects of temperature, strain, and powder-averaging are included, the
resulting structure factor for a single Bragg peak is
S(Q) = Ah (Q) exp[-( 02 (3-45)
mlw e F(Q)2 exp(-Q2)
where Ag(Q) = Ao ,) QQ
4[2(Q) QQ%
and o(Q) =[ o + ( oQ)2 '

The application of these results to the block copolymer SAXS spectra will be discussed
in Chapter 6 following a review of the block copolymer literature in Chapter 4 and a
discussion of the experimental apparatus in Chapter 5.











CHAPTER 4
BACKGROUND AND LITERATURE REVIEW OF BLOCK COPOLYMERS


The polymer molecule is a long flexible chain composed of covalently bonded

repeat units called monomers. Polymeric materials have long been of interest to chemists

and materials scientists due largely to their advantageous mechanical properties and were

only much later recognized as an excellent system for the study of physics. Major

experimental tools that have proved to be extremely useful in recent studies of polymer

structure, morphology, and dynamical behavior include small angle neutron scattering

( SANS ) and synchrotron x-ray diffraction. Outstanding theoretical advancements have

been made, especially through the application of renormalization group ideas to the study

of chain conformations, due primarily to the original work by de Gennes in this field.22


Polymers--General Background


The size of a single polymer molecule is typically described by the

polymerization index, N,
N=- (4-1)
MO
where M is the mass of the entire chain and Mo is the mass of a single monomer.

Generally, a polymer sample is composed of a distribution of chains having varying total

masses. The mass distribution can be characterized by three parameters: the number and
weight average molecular weights, M. and M,, respectively, and the polydispersity

index, n. The number average molecular weight is defined as
IN. M.
Mn =, (4-2)
Mf. XN.








Here Ni is the number of polymer chains of mass, Mi, in the distribution. Similarly, the

weight average molecular weight is given by
Ni, Mi2
M. = (4-3)

The polydispersity is a measure of the width of the mass distribution and is defined to be
the ratio of Mw and Mn. It is easily related to the standard deviation of the distribution,

o, through the equation
S= (< M2 >- < M >2)12 = M, (n-l)l2 (4-4)

Together the above parameters describe the mass distribution of a collection of
polymer chains. The size of individual chains is described by the end-to-end distance, r.
If each monomer is of length I then the rms value of r can be determined by considering a

polymer chain as a random walk containing N steps of length a. It is well known2 that

for an unconstrained or "ideal" random walk the result is

= N P (4-5)
Within this model, the entropy of a single polymer can be calculated and at fixed end to

end distance is
3r2
S(r) = S(0) 2 N (4-6)

From the free energy relation, F=U-TS, it is evident that the chain behaves as a Hooke's

Law spring, when the entropy term dominates the free energy. The effective spring

constant increases with increasing temperature. This illustrates the physical basis for the

contraction of polymer materials which occurs upon heating.

The above calculation is flawed in that chain conformations are allowed in which
the chain "loops back" on itself so that two or more monomers occupy the same physical

space. Calculations which do not allow these conformations are called self avoiding
walks ( SAWs ) and, for dilute polymer solutions, yield a result similar to that for the

ideal random walk:








1 = INv (4-7)

Here v is called the Flory exponent. Theoretical and experimental studies show v=3/s5.s

In the concentrated limit, chains can no longer be considered as independent

random walks and the result, although much more difficult to obtain22 is the same as that

for the ideal random walk, Equation 4-5. Polymer chains are therefore swollen in dilute

solutions containing a good solvent and exhibit ideal ( i. e. random walk ) conformations

in the limit of a concentrated polymer solution or a pure polymer melt.
The radius of gyration, R,, is a parameter conveniently measured in both light

scattering and small angle x-ray scattering experiments and is defined as
Mr? 1/2
Rg I (4-8)


For an ideal random walk the radius of gyration is equal to one sixth of the rms end-to-

end distance.82


The Glass and Melting Transitions


Many polymer systems exhibit a melting temperature, T., above which the

molecules are in constant relative motion. Below Tm, the polymer chains crystallize by

forming helical or layered structures. In many cases, "bulky" monomers prevent efficient

stacking of the polymer chains so that crystallization is prevented. These materials
exhibit a glass transition temperature, Tg. Below T,, relative motion between chains is

hindered by entanglement effects. Often small molecules called plasticizerss" are added
to the material to increase flow by providing lubrication and lowering T,.

Above the glass or melting temperatures and in the absence of chemical

crosslinking, polymer molecules are constantly moving relative to one another. For a

long time, the mode of interdiffusion for polymer molecules was not well understood.








Presently, this process has been clarified and is described by the reputation model

developed by de Gennes.22 Within this model, polymer chains move by "worming" their
way through an open ended tube in which they are partially encased. The mean time

required for a single chain to reptate through a length of tube equivalent to its own length
is called the terminal time, ,, and the following dependence on the polymerization index,

N, has been observed:22
S- NX x = 3 3.3 (4-9)

The translational diffusion coefficient, D,, is given by
2
D N-2 (4-10)

assuming gaussian chain statistics.


Phase Separation in Polymer Blends


Many of the interesting and useful properties of polymer materials result from

segregated structures in polymer mixtures where the component polymers are

incompatible. In this section, the phase separation process in polymer blends is discussed

as an introduction to the more complex block copolymer case.

Many of the features of polymer phase separation are elucidated in a mean field

treatment developed by Flory21 and Huggins.83 Within this model, the free energy of a

mixture of polymers A and B is expressed in terms of the Flory-Huggins interaction
parameter, x:

X=k (EAB-( EAA+EBB) (4-11)

Here Exy is the energy corresponding to an X-Y nearest-neighbor configuration and kBT

is the thermal energy. The polymer system is typically assumed to be incompressible84
so that the volume fractions of A and B monomers can be written: ^A = ( and








(g = 1-0. The free energy for the mixture, including entropy of mixing terms,22 is
I1-1
F=kBTN lnc +kBTNl n(l-0)+X (l-I-) (4-12)

Here and are the A and B chain concentrations, respectively, in dimensionless
NA B

units. It is trivial22 to construct the phase diagram for this system including both the

coexistence and spinodal curves. The result for the symmetric case is very similar in

form to the phase diagram illustrated in Figure 2-1.

From the phase diagram it is evident that at temperatures above the coexistence

curve the equilibrium state of the system is a homogeneous mixture. Below the

coexistence curve, the equilibrium state consists of phase separated A- and B-rich

regions. As described in Chapter 2, many interesting kinetic effects can be observed in a

system exhibiting this type of phase diagram. Specifically, upon lowering the

temperature through the coexistence temperature to a final value in the metastable region,

there exists an interfacial energy associated with formation of A- or B-rich droplets. The

mechanism of transformation is nucleation and growth. When the final temperature is in

the unstable region of Figure 2-1, the interfacial energy disappears and transformation

occurs through spinodal decomposition. However, as mentioned in Chapter 2, a sharp

spinodal line is an artifact of mean field theories such as the Flory-Huggins model and

therefore its true physical significance is unclear.

Nucleation and spinodal decomposition have been observed in studies of phase

separation in blends of polystyrene and poly-( vinyl methyl ether ) by Nishi et al.85 They

used light transmission, optical microscopy, and NMR techniques. They were able to

identify a range for the spinodal point above and below which morphological structures

corresponding to nucleation and spinodal decomposition, respectively, were observed. It

is expected that qualitatively different behavior should be observed for phase separation

in polymer blends in comparison to binary alloys since diffusion in polymer systems is

believed to occur through the reputation mechanism. Reptation yields a chain mobility








which is strongly dependent on the reciprocal space vector, Q, and, as a result, alters the

equations of motion for the system. Nevertheless, Nishi's group found that early time

spinodal decomposition behavior agrees well with Cahn's linearized model of spinodal

decomposition,86 a result which is expected in the presence of long-range interactions,

but never observed in metals. McMaster87 also observed a spinodal point in his light

scattering study of polymer blends. In addition, the coarsening behavior observed

following quenches into the metastable temperature regime, was consistent with the

Lifshitz-Slyozov theory55 described in Chapter 2. However, coarsening following

quenches into the unstable regime occurred much more rapidly. McMaster accounted

for this behavior, invoking a viscous flow mechanism based on models developed by

Tomotika.88

Much work has yet to be done on phase separation in polymer blends, but present

results indicate that studies of these systems, because of their slow molecular diffusion,

can yield valuable information toward understanding kinetic effects.'


Block Copolvmers Bulk Properties


As introduced in Chapter 1, the block copolymer ( BCP) is composed of two or

more chemically distinct polymer chains joined by a covalent bond. When the block

chains are incompatible, phase separation similar to that occurring in polymer blends

takes place. The fundamental properties of bulk block copolymers are summarized in a

review article by Hashimoto et. al.84 As described in Chapter 1, block copolymers
exhibit a dissolution temperature, Td, below which microphase separation transition

( MST ) results in the lamellar, cylindrical, and spherical microstuctures shown in Figure
1-2 and above which the system is homogeneous. The morphology of the microstucture

has been found both experimentally23 and theoretically89 to depend only on the fractional

composition, f, of the block copolymer and is believed to be independent of total








polymerization index, N. In addition, the microphase separated domains are known to

form macrolattices: a lamellar arrangement, a 2-D hexagonal closed packed array of

cylinders and a cubic lattice of spherical domains.

Early theoretical work toward understanding this process was performed by

Meier.90 The driving force for phase separation in a BCP is the repulsive interaction

between the two types of monomers. Since the polymer system is highly

incompressible,28 only those domain configurations which fill all space are allowed. If

the size of the domains exceed the sums of the radii of gyration of the block chains

composing the domain, then that chain must be stretched to satisfy the requirement of

uniform space filling. This results in a loss of conformational entropy and therefore

limits the size of the phase separated domains. There is an additional entropy loss due to

confinement of the junction points (the location of the covalent bond between block

chains) to the interfacial region at the surface of the domain. Meier first identified these

three factors which determine the equilibrium morphology and dimensions of the system.

In addition, Meier recognized that the domains themselves interact through a

potential of entropic origin which in turn is responsible for the macrolattices which the

microdomains are known to form. The physical basis for this interaction can be

understood in the following way. Consider an AB diblock copolymer which forms

spherical domains of the A block in a B matrix. If two A spheres are too far apart, then a

density deficiency is created between them which must be filled by stretching B chains.

In this case, the free energy is increased due to the loss of conformational entropy of the

stretched B chains. Similarly, if two A domains are too close, the volume available to

the B chains between them is too small, again resulting in an increased free energy due to

the loss of conformational entropy. As discussed earlier in this chapter, the contribution

to the free energy from stretching or compressing a polymer chain is quasi-elastic in

form.









One of the first attempts to predict the dimensions of the equilibrium structure is

that of Helfand and Wasserman.91 Within their model, the structure is composed of

chains individually constructed from self avoiding random walks confined to the

appropriate domain space. In the limit of strong segregation for the lamellar

morphology, they found the following scaling relation between the polymerization index,

N, and the microdomain periodicity, d:
d N, 0 = 0.636 (4-13)

Similar results for the value of 0 were obtained for the cylindrical and spherical

morphologies.9293

The result given in Equation 4-13 was supported by studies of the polystyrene-

polyisoprene system by Hashimoto et al.94,95 They cast polymer films from solutions

having fractional compositions that yield the lamellar and spherical structures at varying

total degrees of polymerization. The films were studied with the small angle x-ray

scattering technique. From the SAXS profiles, the periodicities of the lamellar

structures, D, were calculated as well as the lattice constants and spherical radii, R, of

the spherical structures. In addition, Hashimoto's group studied the systematic deviations

from Porod's Law in the high Q region of the scattered profiles ( discussed in Chapter 3)

to estimate the thickness of the interfacial region, 4, separating polystyrene and

polyisoprene domains. Their results indicate the following scaling relations for the

lamellar samples:
d Na (4-14)
S-N

For the spherical structure, the same relations were found to hold and, in addition, the

following empirical relationships were observed:
R N4 (4-15)

p,- N'13








In these equations, pj is the density of junction points on the surface of the spherical

microdomains calculated from the sphere radius, the bulk density of polystyrene, and the

molecular weight of a single polystyrene block. For both morphologies, the periodicity

scales with molecular weight in fair agreement with theoretical predictions.93

The exponent in this relation is greater than the exponent of 1/2 predicted for the
radius of gyration, R, of the chain in a homogeneous melt.22 This indicates that the

individual chains are stretched in the direction perpendicular to the domain surface. A

more recent study confirms this assertion. Hasegawa et. al.9 studied lamellar

polystyrene-polyisoprene using small angle neutron scattering ( SANS ). By deuterating

the polystyrene block of the polymer, they were able to determine the radius of gyration

of these chains from the SANS profile. In their solvent cast films, the lamellae

preferentially lie parallel to the film surface. By scattering from the film edge and
surface they could measure R, both parallel and perpendicular to the domain surface,

respectively. Their results indicate an elongation of the chains by 60% perpendicular to

the lamellar surface and a contraction of 30% in each of the parallel directions.

A vast amount of work on similar systems also supports these scaling relations for

lamellar,25.97,98.99. '0101 cylindrical, and sphericallo0lo02z,03,104.l'1 microdomain

morphologies in the strong segregation limit.

Leibler89 presented a Landau mean field35 analysis of the MST occurring in an

AB diblock copolymer having a fractional composition f, a Flory-Huggins interaction
parameter X, and a total polymerization index N. The order parameter Nf(r), was defined
to be the local density deviation of the A monomer, WA(r), from its mean value in the

homogeneous state; W(r) = < yA(r) f >. The static structure factor, S(Q), is defined as


S(Q) = f exp( -i Q-r) < AA(r) AA(O) > d3r


(4-16)








Leibler expands the free energy in powers of the order parameter. Using the random
phase approximation,22 S(Q) and the free energy are evaluated to fourth order in W(Q)

( the fourier transform of V(i) ). By assuming that the minimized free energy near the

transition is dominated by fluctuations with Q = Q* (Q* Rg- ), Leibler is able to

evaluate the coexistence and spinodal curves in the (XN) f plane and determine the

limits of stability of the different polymer morphologies.

The stable morphology is found to depend not only on the fractional composition,

f (as shown in figure 1-2 ), but also on the temperature. Such transitions, i. e.. from

spherical to cylindrical to lamellar, with changing temperature were later observed

experimentally.23 In addition, Leibler argues that the stable macrolattice for the

cylindrical morphology is the two dimensional hexagonal close-packed structure (HCP)

and for spherical domains is the BCC structure. There have since been reports of a

variety of different macrolattices ( BCC as well as FCC and SC) observed for the

spherical domains.'06

The above formulation yields a scaling relation between the periodicity, d = Q*-',

and the polymerization index, N,
d N1I (4-17)

for the weak segregation limit (where the density profile is a varies smoothly ), distinct

from the relation derived for the strong segregation limit, where well defined interfaces

exist ( Equations 4-13 and 4-14 ). Recent support has been given to this scaling relation

by Chakrabarti et. al.107 who performed Monte Carlo simulations using a two

dimensional lattice model composed of restricted self-avoiding random walks. Their

results and experimental measurements by Shibayama et al.108 also yield the relation

given in Equation 4-17.

Ohta and Kawasaki'09 extended Leibler's theory by including the effects of the

long-range elastic interaction. Using the random phase approximation in the strong









segregation limit they obtain results similar to those of Leibler, and calculate the correct

scaling relation for this limit (Equation 4-13).
More recently, Frederickson and Helfand10 have attempted to improve upon

Leibler's theory by including the effects of concentration fluctuations in the self-

consistent Hartree approximation."1 Their results are similar to those of Leibler, but
they find the transition temperature to be no longer linear in N, but proportional to N1+I,

where (3 is small. The differences between these predictions and those of Leibler are,

however, small and have not yet been resolved experimentally.

In addition to the three commonly observed morphologies: lamellar, cylindrical,

and spherical, a new morphology called the ordered bicontinuous double-diamond

(OBDD) structure has recently been observed23-27 in the polystyrene-polyisoprene

diblock copolymer. The structure is composed of tetrapod domains probably residing on

the double-diamond lattice. This structure was observed in SAXS and electron

microscopy studies when the volume fraction of polystyrene is in the narrow range

between 0.62 and 0.66. The OBDD morphology appears between the cylindrical and

lamellar morphologies in the accepted morphology scheme( Figure 1-2). This discovery

shows the possibility of finding more interesting and technologically useful

morphologies.


Bulk Block Copolymers Kinetic Studies


The theory of the kinetics of the MST is not well developed at the present time.

Kawasaki and Sekimotol12 present a complex model of this process within the reputation

model of polymer diffusion. These results are, however, still very preliminary.

Oono and Bahiana'13 employed a cell dynamical model to describe the MST in a

symmetric diblock copolymer in two dimensions. The partial differential equation used

to describe the time development of the order parameter was a modification of the Cahn-









Hilliard equation.' The two dimensional patterns resulting from their computations

accurately reproduced those observed in electron microscopy studies.84 Using the results
of their computations, dimensional analysis, and comparisons to equilibrium theories
89.109 they obtain a relation between the exponent, 0, describing the equilibrium

periodicity of the microphase structure (Equation 4-13) and the exponent, 4, describing

growth of the microdomains during spinodal decomposition:
L-to (4-18)

That relation is
8 = 2) (4-19)
This implies that in the strong segregation limit, where 0 is believed to be 2/3 for the

lamellar morphology, f = 1/3, in agreement with the Lifshitz-Slyozov result described in

Chapter 2.


Block Copolymer Solutions


The first observation of an ordered structure in block copolymer solutions

occurred in 1966 by Vanzo.114 In studies of the optical reflectance of diblock copolymer

solutions as a function of increasing polymer concentration, C, a critical concentration,

C*, was found at which point a sharp increase in reflectance was observed. In addition,
above C the peak in optical reflectance moved to larger wavelength with increasing C.

From these data Vanzo postulated the presence of an ordered structure at concentrations

above C with an identity period that increases with increasing polymer content.

Early work by Sadron and Gallot25 helped to clarify these effects. They studied a

polystyrene-polybutadiene diblock copolymer mixed with styrene monomer, a selective
solvent ( one in which only the styrene blocks are soluble). After the solutions were

prepared they were illuminated with UV light to polymerize the styrene monomers.

Through this "post-polymerization" technique the polymer solutions are converted into a








gel for electron microscopy studies presumably without altering the microstructure of the

solution. The resulting sample were stained with osmium tetroxide and studied with

transmission electron microscopy (TEM). From these experiments, Sadron and Gallot
identified two critical concentrations, C1" < C"'. Below C,*, the mixture was

homogeneous. In the region between C1" and C2*, cylindrical polybutadiene domains

were observed in a disordered arrangement. Above C2*, the cylindrical domains were

observed in and ordered arrangement. From these results, it was inferred that C2*

corresponds to the quantity C* observed by Vanzo.
Sadron and Gallot25 and Skoulios97 also observed morphological transitions

(i. e. from spherical to cylindrical, or cylindrical to lamellar microstructures ) with

changing polymer concentration in solutions. Using the same "post-polymerization"

technique they were also able to observe morphological transformations upon altering the

fractional composition, f, of the polymer chain.

Pico and Williamsn15 studied the ordering transition in triblock copolymer

solutions using theological measurements. A transition from non-Newtonian to

Newtonian viscous flow was observed upon increasing the polymer concentration above

the measured critical value.

The properties of BCP solutions have been systematically explored by many

researchers, 03, 116.117 using a variety of techniques, but perhaps the most complete

description of these complex systems is presented in a series of five papers by Hashimoto

et al.28, 8,119,120.121 They performed theological and small angle scattering measurements

of the equilibrium structure of diblock copolymers in both selective and nonselective

solvents as a function of temperature and polymer concentration. A brief discussion of

the equilibrium and kinetic results reported by Hashimoto et. al. is given here.








Equilibrium Measurements


The first two papers of this series28 118 report studies of the diblock copolymer

polystyrene (PS )-polybutadiene (PB ) in the selective solvent n-tetradecane (C14)
which dissolves the PB block. The BCP had a number average molecular weight, M. =

5.2 x 104 and a fractional composition, f = 0.30. The solutions studied had the following

polymer concentrations: 8, 11, 20, 35, and 60 wt.% polymer. At these concentrations

and at this fractional composition the MST results in spherical PS domains within a
PB/C14 matrix. X-ray scattering profiles were acquired at varying temperatures upon

heating and cooling. The x-ray spectra were corrected for parasitic scattering and

collimation errors resulting from instrumental resolution.

In these experiments, Bragg peaks were evident in the low temperature spectra for

all of the solutions in the relative locations Nl/, "f2, '3, and v4 after collimation

correction. The lattice constants were determined from the first Bragg peak location and

the average radius of the PS spheres was determined from the location of the first

maximum in the spherical form factor as described in Chapter 3. It was speculated that

the macrolattice of PS domains was simple cubic ( SC ) rather than body-centered cubic

( BCC) from comparisons of the calculated and stoichiometric PS volume fractions for

these structures. This type of calculation is described in detail in Chapter 6.

As the temperature, T, for each solution was increased, the Bragg peak amplitude

decreased continuously while the full width at half maximum ( FWHM ) remained

unchanged. In addition, the calculated lattice constants decreased slowly with increasing
temperature. At a polymer concentration dependent temperature To, the Bragg amplitude

decreased abruptly, accompanied by an increase in the FWHM and a drop in the lattice
constant. As the temperature was increased further above T,, the peak amplitude and

lattice constant continued to decrease, while the width increased rapidly. Finally above
T = Td, no further evidence of the peak structure remained.








Hashimoto et al. associated the temperature To with the disordering of the

macrolattice resulting in a "liquid-like" arrangement of the PS spheres. The temperature,
Td, is identified as the dissolution temperature at which point the PS domains totally

dissolve. In order to clarify these interpretation, theological measurements were made on

the polymer solutions as a function of temperature. Two transitions in the mechanical
properties were identified. Upon raising T above To, a transition from non-linear plastic

flow to linear non-Newtonian flow was observed in support of their identification of T,.

Also, at T = Td, a transformation from linear non-Newtonian flow to linear Newtonian

flow was seen in agreement with earlier studies15 identifying this behavior with the

dissolution point.

The authors identify a number of quantitative trends in the thermal behavior of

the system. The dependence of the apparent lattice spacing, a, on temperature both

above and below the ordering point were consistent with the following empirical relation
above and below the discontinuity at T,:

a T- (4-20)

The discontinuity in a at T = To is thought to be due to the change in structure occurring

at this temperature. Above T,, the peak is no longer a Bragg peak, and therefore,

determining a lattice constant from the peak position is erroneous. The decreasing trend

of a with T is the result of two factors. First, the PB chains between the domains

contract upon heating to increase their conformational entropy. Second, at higher
temperatures the effective interaction, given by the Flory-Huggins parameter, X 1/kBT,

is reduced resulting in greater intermixing of the PS and PB chains which in turn causes a

contraction of the lattice. The authors believe the second of these two is the more

important effect at all temperatures.
The Bragg widths below To appear to be independent of temperature. The

authors offer an interesting explanation for this result. The domain-domain interaction in

BCP solutions is assumed to be much the same as described by Meier for the pure








polymer system, being quasi-elastic. The change in free energy, AF, due to small

elongations, Aa, of the lattice constant from its equilibrium value can therefore be

expressed as
AF f(C, T) (Aa)2 (4-21)

Here f(C,T) the spring constant of the PB chains, is a function of the polymer

concentration, C, and temperature. The authors argue that if the distribution of strains

satisfies the Boltzmann distribution, then the distribution of lattice constants is
characterized by a gaussian form having a width parameter, o,,
kBT
a2 (CT)-f(CT) (4-22)

The "spring constant" for the PB chains is entropic in origin as described earlier in this

chapter so that
f ( C, T) kBT (4-23)

Together 4-22 and 4-23 imply a strain distribution which is independent of temperature.

Therefore the Bragg peak width is expected to be independent of temperature when strain

is the dominant contributor to the width.
The authors propose the following intuitive explanation for disordering of the

macrolattice. As the temperature is increased and there is greater intermixing of the PS

and PB chains, the PS blocks extend further into the interstitial region between the

spheres. The PB chains are thus relaxed by intermixing and the effective potential well

of equation 4-21 becomes more shallow until melting of the macrolattice results. When

the demixing becomes pronounced, the domain-domain interaction is no longer purely

entropic. The energetic contribution to the interaction is claimed to be the cause of the

increasing trend in the peak width with temperature just below the dissolution

temperature.

Hashimoto et al. also observed several trends with increasing polymer
concentration. Both To and Td increased with increasing concentration due to reduced








screening of the PS-PB interaction which results in a higher Flory-Huggins interaction

parameter, X. The measured lattice constants were consistent with the scaling relation
a C-'" (4-24)

Again, the data extend significantly less than a decade in both a and C. The PS sphere

radius, R, was seen to increase slowly with increasing C. This trend was said to be the

result of a shifting in the relative weights of the energetic and entropic terms in the free

energy. At higher concentrations, more chains are added to each domain increasing the

sphere radius. As a result the PS chains are stretched and some conformational entropy

is lost, but the interaction energy from the surface to volume ratio is reduced.


Time-Dependent Experiments


As an extension of their earlier work on equilibrium properties, Hashimoto et. al.

performed what appear to be the only kinetic studies of the block copolymer system to

date.122,123.124 They again studied PS PB diblock copolymer in the selective solvent C14.
The fractional composition was f = 0.30, and the molecular weight was Mn = 5.2 x 104.

The polymer concentrations studied were 20, 25, 30, 35, and 40 wt.% polymer in

solvent. They attempted to measure the polymer diffusion coefficient as a function of

temperature by performing temperature jump experiments using the time-dependent

small angle x-ray scattering technique. These measurements are described in some detail

to provide a comparison with the experiments reported in this thesis.

The polymer solutions were initially held in equilibrium at room temperature.

The temperature was then rapidly raised to a point above the dissolution point. With the

thermodynamic driving force for the MST removed, it was thought that the system would

relax to the homogeneous state through diffusion of the center of mass positions of the

diblock chains. By studying the SAXS profile as a function of time, it was their goal to
measure the diffusion constant, DC, for the diblock chains in solution.








In fact, even at elevated temperatures, the interaction between PS and PB

monomers will still affect the dissolution process. In addition, the diffusivity of the

whole chain is a function of the diffusivities for each of the individual blocks. As a
result, the measured quantity is an effective diffusivity, De.

The polymer solutions were placed in an aluminum sample cell, which was sealed

with mica windows, and held at room temperature. The sample cell was then rapidly

transferred manually to a large copper heating block which was controlled at the desired

final temperature. The sample temperature exhibited a sigmoidal variation with time and

a time constant of 6 seconds. Placement of the cell in the heating block triggered the

start of data acquisition. Sixty-four consecutive scattered spectra were taken in real time

following the temperature jump, each with an exposure time of 2 seconds. Due to poor

counting statistics, temperature jumps to each final temperature were repeated 10 times

and for each iteration, the spectra corresponding to the same time slice were summed.

The resulting SAXS profiles exhibit a first-order peak which decays rapidly with

time following the temperature jump. A relation between the scattered intensity, I(Q,t),
and Df. was derived involving three assumptions: an isotropic particle distribution,

Fickian diffusion of the diblock chains, and no center of mass motion for the

microdomains while they dissolve, i. e.. no macrolattice disordering. Their result is

I(Q,t) = I(Q,t=0) e-2 Q2 Dl t (4-25)
From this expression, the intensity decays exponentially with time and so D.f can be

determined from the slope of the semi-log plot.

The interpretation of these results is complicated by the speed with which

transformation takes place. Much of the transition occurs non-isothermally. In order to

account for these non-isothermal effects, a crude model for the time development of the

scattered intensity was derived and is described by the following recursion relation:

I(Q,til) = [ I(Q,) I(Q,T) ] e-2 Q2 Ddf(T) At + I(Q,Tj) (4-26)








Here the temperature jump is divided into time increments, At. The intensity at time, tj.

is determined from the intensity at the previous time increment, tj. During each

increment, the intensity moves toward the equilibrium intensity distribution, Ie

corresponding to the temperature Tj which is the temperature of the sample at time tj.

In order to further simplify the problem, D, is assumed to follow an Arrhenius

temperature dependence:
Dff = Do exp (-AH ) (4-27)

Their model for the temporal development of the intensity at any scattering vector, Q, is
a function of the two parameters, Do, and the activation energy, AH,.

The authors plot the logarithm of the scattered intensity at a specific scattering

vector, Q*, near the first-order maximum as a function of time following the quench.

The data are generally non-exponential at early times becoming exponential at later

times. This behavior is accurately reproduced in their model described above. The

authors present two methods of extracting the desired parameters from the data. First,
the diffusion constants Def are determined from the late time exponential region of the

data and from a plot of Dff versus temperature, AH, and Do can be calculated. The

second method, involves varying Do and AH. and individually matching the model

( Equation 4-26 ) to the measured data. The two methods gave similar numerical results
for AHa and Do. The data were found to be consistent with the following relation:

Do- C'.75 (4-28)

Results for the activation energy, AHa, show no consistent trend, but yield results for all

concentrations in the vicinity of 10 kcal / mole.

As can be seen from this brief discussion of the current literature, experimental

studies of BCP transition phenomena are still quite limited. These systems are rich in

their diverse microstructural morphologies and the transitions which occur between them





78


and the homogeneous melt. This complex behavior has been the stimulus for the series

of experiments reported here.













CHAPTER 5
THE POLYMER XRAY SCATTERING EXPERIMENT


In this chapter, the experimental apparatus is described beginning with a brief

overview of the data acquisition system. This is followed by a more detailed discussion

of the major components of the apparatus: the polymer x-ray scattering furnace and the

SAXS apparatus. The chapter is concluded with a discussion the experimental

procedures employed in these studies.


System Overview


The objective of these studies was to observe structural changes in block

copolymer solutions as a function of temperature and time by measuring the SAXS

profile. The data acquisition system which was devised for this purpose is shown in

Figure 5-1. The control system is centered around an IBM PC AT operating in the

ASYST environment. The programs and routines written to control the data acquisition

system are described in Appendix A. The computer controls the x-ray detector and

polymer x-ray scattering furnace through four devices controlled via the General Purpose

Interface Bus. These devices are a PAR 1461 detector controller, a Stanford Research

Systems Model DG535 digital delay generator, a Micristar temperature controller, and a

Keithley 197 digital multimeter.

The PAR detector interface controls the PAR 1412 XR position sensitive x-ray

detector. Upon receipt of commands from the computer, the detector interface relays

those commands to the detector, then stores up to 512 acquired spectra for DMA transfer






















































A schematic of the data acquisition system.


Figure 5-1









to the computer at a later time. The silicon diode array detector is evacuated using a

Leibold D8A two-stage roughing pump and requires a cooling system ( A Fisher Model

900 Circulator ). The details of the SAXS apparatus including the detection system are

discussed momentarily.

The sample temperature is monitored and controlled by a Micristar temperature

controller operating in the proportional mode of control. In this mode, the Micristar

compares the sample temperature to the programmed setpoint and outputs a DC voltage

between 0 and 5V which is proportional to the difference. This voltage is sent to a

specially designed heating circuit ( described in Appendix B ) which in turn controls the

current to heating resistors in the polymer x-ray scattering furnace. In this way, the

sample temperature is controlled with a stability of +- 0.3 oC. The details of the x-ray

scattering furnace are discussed momentarily. The sample temperature as determined

from the thermocouple voltage emitted by a thermocouple embedded in the sample is

also monitored by a Linseis Model L4100 chart recorder and the Keithley multimeter.

The time-dependent measurements were performed by annealing the BCP
solution above its dissolution point, Td, and then rapidly lowering the temperature to a

fixed point below Td and observing the structural transition through the SAXS profile.

This thermal quench is achieved by forcing a fixed amount of coolant rapidly through a

quenching channel in the sample mount. The measured coolant was injected into the

aperture of the quenching channel inlet to the furnace. Polypropylene tubing extending

from a cannister of helium was then connected to the quenching channel inlet. A

solenoid valve placed between the furnace and the helium was used to control the time

and duration of the helium burst which was initiates the quench. The solenoid valve is

controlled by the digital delay generator through a specially designed solenoid interface

circuit, a schematic of which is shown in Appendix C.








A description of the procedures performed in these experiments is given later in

this chapter. Now, a more detailed discussion of the major components to the data

acquisition system, the polymer furnace and the SAXS apparatus, is presented.


The Polymer X-ray Scattering Furnace


The polymer quenching furnace is composed of three parts : the furnace body, the

translational stage and the sample mount. The furnace body and the translational stage

were originally designed for use in other experiments and were used here with relatively

minor modifications. The sample mount, however, was specifically designed for these

experiments and will therefore be described in slightly more detail.


The Furnace Body


The body of the furnace is essentially a vacuum chamber composed of a stainless

steel base and an aluminum lid. These two parts are joined with an o-ring seal and a

number of bolts arranged in an octagonal pattern. The furnace base as seen from above

is shown in Figure 5-2.

The lid is a cylindrical aluminum shell with two aluminum windows for entrance

of the incident x-ray beam and exit of the scattered radiation. The windows are
aluminum plates with 21/4" diameter holes. The window frames are bolted to the lid for

easy replacement. The frames are sealed to the lid with o-rings. A number of window

materials were tried and the two optimal materials were beryllium and Kapton. Kapton

is the tradename for a polyimide material commonly distributed in sheets and used as

substrates or vacuum windows. Beryllium is the standard window material used in wide

angle x-ray scattering and offers the best percentage transmission ( 96.17% for a 5 mil

sheet). However, it is a horrendous parasitic scatterer in the region of interest in these



















































A top view of the polymer x-ray scattering furnace illustrating the relative
placement of its various features described in the text.


Figure 5-2









experiments. Parasitic scattering is defined as that unwanted component to the scattering

profile which is a result of scattering from slits and window materials. This quantity was

measured as the integrated number of counts per second above background. This

quantity for the 5 mil beryllium sheet was 8.0 times larger than for the 5 mil Kapton

sheet in the 0.05 to 1.0 degree range and 5.2 times larger in the 0.05 to 0.35 degree

range. Although the percentage transmission was slightly less (transmission = 91.13%)

for the Kapton, the reduction in parasitic scattering was much more important. The

Kapton windows were sealed to the window frame with Omegabond epoxy and resin.

The chamber was evacuated with an Edwards ETPG pumping station which

incorporated a roughing pump and a turbomolecular pump. The ultimate pressure

typically achieved under experimental conditions was between 1 and 10 millitorr. The

pressure was much lower ( below the level which could be read from the Varian

thermocouple gauge controller ) when the windows were replaced with solid aluminum

plates. For this reason, it was concluded that the ultimate pressure was being limited by

permeation through the Kapton windows. The vacuum pressure probably could have

been improved with thicker Kapton windows, but the pressure was satisfactory for these

measurements and further improvement was unnecessary.

The furnace body has a number of vacuum feedthroughs. These are shown in

figure 5-2 and include the following. An eight-pin electrical feed through is used to pass

current to the resistors which heat the sample. The feed through was a standard high

vacuum flange which bolted to the side of the furnace and was sealed with teflon tape.
Two 1/4" outer diameter, I/g" inner diameter copper tubes were passed through the

furnace wall and were sealed with standard Swadgelock fittings. These tubes were used

to pass coolant through the sample mount to initiate the thermal quench during the

kinetic measurements. One feedthrough passes four thermocouple connections into the

chamber: two chrome and two alumel. One chromel and one alumel connection are

used to form the thermocouple junction which compose a single thermal sensor. Both








sets of connections were used, one to monitor the sample temperature and one to monitor

the sample mount temperature. A Varian Type 0531 thermocouple gauge head was

attached in the location shown in figure 5-2 with teflon tape. The sensor was connected

to a Varian 843 gauge controller.
The chamber was evacuated through the 1/2 vacuum port shown in the Figure.

Two additional high vacuum flanges are available on the scattering furnace, but are not

being used at this time.


The Translational Stage


The base of the furnace body is cut into a dovetail shape. The base mates with a

brass translational stage, the bottom of which is itself cut into the dovetail shape in the

orthogonal direction. An additional brass plate mates with this piece, and by sliding the

dovetail within its track, two independent translations are provided for centering the

sample in the x-ray beam. The translational stages are mounted on a Huber Four Circle

Diffractometer ( the four independent orientation angles were a little overkill in a small

angle scattering experiment ) via the standard Huber Goniometer connecting ring. The

connecting ring can be translated vertically. Along with those translations allowed by the

brass translational stage, three degrees of freedom are provided for centering the sample

in the beam. The connecting ring is typically adequate for small samples or sample

chambers, but is inadequate for a furnace of this size. For this reason, appropriate

spacers were chosen and the vertical translation of the connecting ring was set so that the

furnace bottom rested on the spacers. Then four additional screws were used to further

secure the furnace to the diffractometer.

As stated previously, the furnace body and translational stage were originally
designed for use in the Cu3Au experiments being performed in the laboratory and were

used with minor modifications (i. e. installation of the quenching loop feedthoughs ).








However, the sample holder was designed specifically for these experiments. Many

iterations were required to optimize the performance of the sample mount. The

development of the sample mount is described in Appendix D. Only the final version is

described here.


The Sample Mount


The sample mount is composed of two parts, the heating unit and the sample cell

both of which are shown in Figure 5-3. the heating unit was mounted in the furnace and

remained there throughout the experiments, while the sample cell was replaced for each

polymer solution.

The sample cell is a 27 mm x 12 mm x 3 mm copper block. A slot of dimensions

2 mm x 5 mm was cut in the copper block to serve as the sample cavity. The polymer

solutions were held in this cavity by two 2 mil Kapton windows. The Kapton windows

were sealed using Omegabond high temperature epoxy. Two 1/16" stainless steel masks

were bolted to the front and back of the sample cell to provide structural support to the

seal. these masks were found to be essential in preserving the integrity of the seal at the

elevated temperatures used in these experiments.

The sample temperature was measured by inserting an Omegaclad K-type

thermocouple directly into the sample cavity through a 1/16" diameter hole. these

thermocouples are composed of one 3 mil chromel and one 3 mil alumel wire extended

parallel to each other and surrounded by a cylindrical stainless steel shell. This

cylindrical sheath is packed with ceramic powder which prevents contact between either

of the alloy wires and the outer shell. The metal coating and ceramic powder are sheared

off at both ends of a 2" long section. At one end, the leads are welded together to form

the thermocouple junction. at the other end, the wires are connected to the thermocouple


















































The final sample mount is illustrated in a front and top view. Evident in
this drawing are the quenching tubes, the heating resistors, the
thermocouple, the stainless steel masks which support the Kapton
windows, and the stainless steel brace which firmly clamps the sample
cell into the sample mount.


Figure 5-3